Investigating the impact of a Realistic Mathematics ... - Main... · of a Realistic Mathematics...
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mmu.ac.uk/education Manchester Metropolitan University Investigating the impact of a Realistic Mathematics Education approach on achievement and attitudes in Post-16 GCSE resit classes Sue Hough, Yvette Solomon, Paul Dickinson and Steve Gough
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mmu.ac.uk/education
Manchester Metropolitan University
Investigating the impact of a Realistic Mathematics Education approach on achievement and attitudes in Post-16 GCSE resit classesSue Hough, Yvette Solomon, Paul Dickinson and Steve Gough
We are grateful to the Nuffield Foundation for their funding and support.
The Nuffield Foundation is an endowed charitable trust that aims to improve social well-being in the widest sense. It funds research and innovation in education and social policy and also works to build capacity in education, science and social science research. The Nuffield Foundation has funded this project, but the views expressed are those of the authors and not necessarily those of the Foundation. More information is available at www.nuffieldfoundation.org
Acknowledgements
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Contents
Contents ..................................................................................................................................... 1
List of Figures ............................................................................................................................. 2
List of Tables .............................................................................................................................. 3
Acknowledgements .................................................................................................................... 4
Summary .................................................................................................................................... 5
Key findings and recommendations .......................................................................................... 6
1 Introduction ........................................................................................................................ 8
1.1 The post-16 landscape ............................................................................................... 8
1.2 Mathematics teaching in England: pressure towards early formalisation ................ 9
1.3 The challenge of GCSE resit ..................................................................................... 11
1.4 A note on the GCSE context of this project, and the new GCSE specification ........ 11
2 The Realistic Mathematics Education approach .............................................................. 12
2.1 The background to Realistic Mathematics Education ............................................. 12
2.2 Use of context .......................................................................................................... 12
2.3 Use of models and two ways of ‘mathematising’ .................................................... 13
2.4 Multiple strategies and formalisation: redefining ‘progress’ .................................. 13
2.5 Applying RME design principles for GCSE resit ........................................................ 15
3 The GCSE re-sit project ..................................................................................................... 17
3.1 Research questions .................................................................................................. 17
3.2 Methodology ............................................................................................................ 18
4 Research findings: the impact of RME.............................................................................. 23
4.1 Overall test performance and script analysis of the Number test .......................... 23
4.2 The impact of the RME approach on classroom interaction ................................... 44
4.3 The student experience: GCSE resit classes and RME ............................................. 64
4.4 The teachers’ view ................................................................................................... 79
5 What have we learned? .................................................................................................... 93
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5.1 The post-16 landscape revisited .............................................................................. 93
5.2 Suitability of the RME approach for GCSE resit ....................................................... 94
6 Implications and recommendations ................................................................................. 98
7 References ...................................................................................................................... 101
8 Appendices ..................................................................................................................... 106
8.1 Appendix 1 – The pilot study, 2012-2013 .............................................................. 107
8.2 Appendix 2 – Design principles .............................................................................. 109
8.3 Appendix 3 – The number and algebra module contents ..................................... 116
8.4 Appendix 4 - The survey bar lesson ....................................................................... 118
8.5 Appendix 5 – The independent evaluation report ................................................ 120
List of Figures
Figure 2-1 The tip of the iceberg, taken from Webb, Boswinkel and Dekker (2008) ............................................ 14
Figure 2-2 Progression from ‘model of’ to ‘model for’ .......................................................................................... 15
Figure 2-3 Early section of material in the survey bar lesson ............................................................................... 15
Figure 4-1 Student A, photocopier question, pre-test attempt ............................................................................. 27
Figure 4-2 Student A photocopier question, post-test attempt ........................................................................... 28
Figure 4-3 Student B, photocopier question, pre-test attempt ............................................................................. 29
Figure 4-4 Student B, photocopier question, post-test attempt ........................................................................... 29
Figure 4-5 Student C’s ratio table strategy for the photocopier question ............................................................ 30
Figure 4-6 Student D’s ratio table strategy for the photocopier question ............................................................ 30
Figure 4-7 Student E’s ratio table strategy for the photocopier question............................................................ 31
Figure 4-8 Student D’s pre-test attempt to compare two speeds. ........................................................................ 32
Figure 4-9 Student D’s post-test attempt to compare two speeds. ...................................................................... 32
Figure 4-10 Student F, ratio question, pre-test attempt ....................................................................................... 35
Figure 4-11 Student F, ratio question, post-test attempt ..................................................................................... 35
Figure 4-12 Student G, ratio question, pre-test attempt ...................................................................................... 36
Figure 4-13 Student G, ratio question, post-test attempt..................................................................................... 36
Figure 4-14 Student H, ratio question, pre-test attempt ...................................................................................... 37
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Figure 4-15 Student H, ratio question, post-test attempt..................................................................................... 37
Figure 4-16 Student images for cutting a pizza into 9 equal slices: example 1 .................................................... 39
Figure 4-17 Student images for cutting a pizza into 9 equal slices: example 2 .................................................... 39
Figure 4-18 Using a bar model representation to find 5/8 of £600 ...................................................................... 40
Figure 4-19 Using a bar model representation to solve a reverse percentage question ...................................... 41
Figure 4-20 Misapplication of the halving strategy .............................................................................................. 42
Figure 4-21 Classroom turn-taking structure (From Ingram & Elliot, 2016) ......................................................... 46
Figure 4-22 Sarah's solution ................................................................................................................................. 48
Figure 5-1 Algebra post-test example of a student using an informal strategy to solve an equation .................. 96
Figure 8-1 Introducing the context ..................................................................................................................... 109
Figure 8-2 Introduction of fair sharing ................................................................................................................ 109
Figure 8-3 Introduction of ‘model of’ .................................................................................................................. 110
Figure 8-4 Repetition of ‘model of’ ..................................................................................................................... 110
Figure 8-5 Bridging the gap between informal and formal ................................................................................ 111
Figure 8-6 Closing the gap and the top of the ‘iceberg’ ...................................................................................... 111
Figure 8-7 Use of models in multiple areas in mathematics ............................................................................... 112
Figure 8-8 Visualising fractions to make sense of common demoninator .......................................................... 112
Figure 8-9 Using the bar in a familiar context .................................................................................................... 113
Figure 8-10 The bar as a ‘model of’ .................................................................................................................... 113
Figure 8-11 Recognising the limitations of the bar ............................................................................................. 114
Figure 8-12 Ratio tables as a more flexible model – a ‘model for’ ..................................................................... 114
Figure 8-13 Effect of intervention on attainment in number .............................................................................. 130
Figure 8-14 Effect of intervention on attainment in algebra .............................................................................. 131
List of Tables
Table 1-1 KS5 GCSE mathematics resit entry and achievement based on KS4 achievement (Source, DfE 2016) ... 8
Table 3-1 The host teachers .................................................................................................................................. 19
Table 3-2 The students .......................................................................................................................................... 19
Table 3-3 Phases and testing ................................................................................................................................ 20
Table 4-1 Percentage of marks gained per question by intervention and control groups in Number pre- and
post-tests .............................................................................................................................................................. 24
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Table 4-2 The photocopier question results .......................................................................................................... 26
Table 4-3 Performance in the given ratio question ............................................................................................... 33
Table 4-4 Teachers' use of time in 'reform' and traditional classes as reported in Boaler (2003) ........................ 45
Table 4-5 Teacher use of time in Lesson A ............................................................................................................ 49
Table 4-6 Time, topic, activity and notable features across Lesson A .................................................................. 51
Table 4-7 Teacher use of time in Lesson B ............................................................................................................ 56
Table 4-8 Time, topic, activity and notable features across Lesson B ................................................................... 59
Table 8-1 Pilot results ......................................................................................................................................... 107
Table 8-2 Test absence ....................................................................................................................................... 128
Table 8-3 Descriptive statistics for use of RME approach for number items ...................................................... 132
Table 8-4 Linear regression statistics to show relationship between use of RME and post-test attainment in
number ............................................................................................................................................................... 133
Acknowledgements
Our thanks go to all the students and teachers who allowed us into their classrooms and
gave their time to this project.
We are grateful to the Nuffield Foundation for their funding and support.
The Nuffield Foundation is an endowed charitable trust that aims to improve social well-
being in the widest sense. It funds research and innovation in education and social policy and
also works to build capacity in education, science and social science research. The Nuffield
Foundation has funded this project, but the views expressed are those of the authors and not
necessarily those of the Foundation. More information is available at
www.nuffieldfoundation.org
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Summary
Students who have not achieved an acceptable pass grade in GCSE Mathematics by the age
of 16 (a grade C at the time of this project) are required to work towards this as part of a 16-
19 study programme. As students with a history of repeated failure in mathematics, they
have low confidence levels and a tendency to rely on mis-remembered rules applied
without understanding. They are often disengaged from mathematics and many do not see
it as relevant beyond a pass grade requirement for entry to further training or jobs. Re-sit
examination success rates are often poor. The short duration of post-16 resit courses, from
September to May, means that teachers face a number of challenges: they need to build
students’ confidence and improve achievement, but they do not have time to address gaps
in knowledge while targeting the required curriculum coverage.
This project trialled the use of an alternative approach: Realistic Mathematics Education
(RME) prioritises use of context and model-building to engage and motivate students,
enabling them to visualise mathematical processes and make sense of what they are doing
without resorting to rules and procedures which have no meaning. Using materials based on
RME design principles, the research team developed and delivered two short modules, on
Number and Algebra, employing a quasi-experimental design to assess impact on
performance in four GCSE resit classes. The main objectives of the project were to assess:
The impact of an intervention based on RME on students’ achievements and attitudes in
Post-16 GCSE resit mathematics;
The potential of an RME-based approach to positively impact on students’
understanding of mathematics and their engagement in class;
The practical issues involved in the adoption of an RME approach with Post-16 GCSE
resit classes;
Teachers’ perceptions of the role of RME in the context of GCSE resit classes;
The implications for a wider application of an RME approach in this context, and
teachers’ professional support needs.
In addition to Number and Algebra pre- and post-tests, a range of quantitative and
qualitative data was collected, including attitude questionnaires, student and teacher
interviews, and classroom observations.
The project was independently evaluated by the Centre for Development and Research,
Sheffield Hallam University (CDARE), who analysed the pre-/post-test comparisons, and
pre/post attitude questionnaires. Their results showed small but significant gains for the
intervention group in the Number module post-tests, although not in Algebra. In addition,
there was a significant difference between intervention and control groups in terms of the
use of RME methods to answer questions. There was no discernible impact on attitudes.
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The Manchester Metropolitan University research team carried out a range of qualitative
analyses to assess the impact of the intervention. Qualitative script analysis of Number post-
tests suggested that impact of the RME intervention was significant, not only in terms of
usage, but also in terms of enabling students to make progress. Analysis of the impact of
RME on classroom interaction showed that RME has the potential to change classroom
norms and encourage students to engage in discussion and meaning-making, despite some
resistance to the approach as evidenced in interviews with both students and teachers. In
the case of the Algebra module, various factors, notably time pressure, may have led to lack
of gains in this topic. GCSE work requires that students reach some demanding formal
levels of symbolisation and algebraic manipulation, and in some cases, the gradient of
moving from the RME contexts written into the materials through to answering GCSE-type
questions was too steep in the time available. Despite the challenges of bringing about
change in this difficult context, both students and teachers expressed positive views about
the ability of the RME approach to enhance understanding.
The project findings lead to recommendations for extending the RME intervention within
one year courses, or extending to two year courses, and focusing on changing classroom
cultures and improving students’ learning skills in order to support new understandings of
mathematics with RME. Recommendations are also made for sustained Continuing
Professional Development (CPD) using lesson study which enhances teachers’ pedagogic
and subject knowledge while enabling them to successfully employ RME design principles to
support students’ developing understanding.
Key findings and recommendations
Key Findings
The independent evaluation found that students receiving a short Realistic Mathematics
Education (RME) intervention showed improved attainment on post-test performance in
Number but not Algebra. There was also a significant difference between intervention
and control groups in terms of the use of RME methods to answer questions.
Qualitative script analysis of Number post-tests revealed the impact of the RME
intervention in terms of:
o use of the bar and ratio table models to find a route to a solution
o allowing some students to re-engage with informal sense making strategies
o providing a structure within which to organise and record thinking
o encouraging flexibility and creativity
Script analysis also revealed that students could recognise the unifying potential of the
models for answering questions across a range of topics.
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Analysis of the impact on classroom interaction showed the potential of RME for
encouraging students to engage in discussion and meaning-making.
Students were responsive to RME but did show some resistance to new methods which
they saw as slow.
While successful in individual lessons, Algebra teaching suffered from time pressure in
the resit context.
Both students and teachers expressed positive views about the ability of the RME
approach to enhance understanding.
Teachers were positive about the RME approach but expressed concerns about pace in
the context of the challenges of teaching GCSE resit.
Recommendations
GCSE mathematics resit courses should ideally be extended to two years and
incorporate an RME approach.
Within one-year courses, an RME-based approach for the whole of term 1 would
establish models and pedagogies on which to build a more effective term 2 and 3 focus
on revision and past papers.
Courses need to focus on changing classroom cultures towards new norms of discussion
involving questioning, sharing and evaluating ideas in order to support new
understandings of mathematics and encourage students to take ownership of the
subject. Courses also need to work on enhancing students’ learning skills by encouraging
them to question their own and others’ strategies.
Teachers need sustained CPD in order to support the development they seek for
addressing the challenges of GCSE resit teaching.
A lesson study model of CPD would enhance teachers’ pedagogic and subject knowledge
while enabling them to successfully employ RME design principles to support students’
developing understanding.
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1 Introduction
1.1 The post-16 landscape
In 2011, the Wolf Report highlighted the importance of GCSE Mathematics and English at
grades A*-C to a young person’s employment and education prospects, noting that:
Less than 50% of students have both at the end of Key Stage 4; and at age 18 the
figure is still below 50%. Only 4% of the cohort achieves this key credential during
their 16-18 education.
This statement of concern reflected a general trend: of the 36.4 percent of young people
failing to gain grades A*- C in their Mathematics GCSE in 2009/10, only 17.6% (of the 36.4%)
went onto re-enter for the examination during their 16 -18 education, less than half of
whom went onto achieve a grade C or above - just 2.7% of the original KS4 cohort (DfE,
2016). The situation has not improved in subsequent years, with DfE (2016) reporting that
although GCSE mathematics entries during 16-18 have increased since 2011/12, “there has
been no increase in the proportion achieving A* to C” (p.1). This is illustrated in the most
recent available statistics, shown in Table 1-1. Students included in Key Stage 5 cohorts are
based on those finishing Key Stage 4 two years before.
End of Key Stage 4 in 2009/10 2010/11 2011/12 2012/13
Percentage of KS4 students not
achieving grade C or above
36.4% 33.0% 29.1% 28.2%
End of Key Stage 5 in 2011/12 2012/13 2013/14 2014/15
Percentage of those leaving KS4
without A*-C entered for GCSE resit
17.6% 17.9% 19.6% 24.6%
Percentage of those leaving KS4
without A*-C who achieved a grade
C or above in KS5
7.4% 7.0% 7.1% 7.1%
Percentage of original cohort 2.7% 2.31% 2.1% 2.0%
Table 1-1 KS5 GCSE mathematics resit entry and achievement based on KS4 achievement (Source, DfE 2016)
However, GCSE Mathematics and English have long been seen by employers as the accepted
standard, and as such are the gateway to many careers both vocational and otherwise. It is
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perhaps unsurprising that in response to the findings of the Wolf report (2011) a national
requirement was introduced in September 2013 that all students aged 16-19 who did not
have Grade C passes in GCSE Mathematics and English must work towards this qualification.
This creates a number of challenges for the Post-16 sector: Firstly, students who have
previously failed to gain an acceptable pass at GCSE, and who in the past would have opted
to study functional mathematics courses, must currently follow a course which requires a
repeat of the same content they previously failed at. Secondly, the subsequent impact on
student numbers means that a significant proportion of non-specialist teachers will now be
required to teach GCSE resit. These challenges are exacerbated by the policy and practice
context of mathematics teaching in England.
1.2 Mathematics teaching in England: pressure towards early formalisation
Mathematics teaching in England is deeply embedded in its evaluation and accountability
systems. The Office for Standards in Education (Ofsted) framework for inspections (Ofsted,
2012) emphasises expectations that pupils will typically make the equivalent of two whole
levels of progress from one Key Stage to the next. Schools are required to evidence this,
leading to the adoption of rigorous pupil tracking systems and regular testing in
mathematics, with interventions for those who are not making the required progress.
At lesson level, the emphasis on student progress within predetermined time limits has
increased the practice of setting lesson objectives and sharing these with students at the
start of a lesson. The National Strategy documentation (DfE, 2013) provides teachers with
lists of objectives, referring to specific mathematical content and focussing on the formal
methods students need to learn. So, for example, the geometry and measures strand of the
latest version of the Key Stage 3 (age 12 – 14) programmes of study states that “Pupils
should be taught to derive and apply formulae to calculate and solve problems involving:
perimeter and area of triangles, parallelograms and trapezia, …” (p. 8). The Key Stage 2
programmes of study require students to “use formulae for area” and “calculate” (p. 43).
Consequently, most teachers in England have adopted the practice of setting lesson
objectives which refer to acquisition of a formal process.
Teachers in England are consequently under pressure to move towards formal mathematics
as quickly as possible, and this has an impact on how they use context to support teaching.
Any contexts which may have been introduced are quickly dropped to allow for abstraction
and development of the desired formal methods. Progression is seen in terms of students’
acquisition of these methods, their ability to use them in more complicated situations (often
‘bigger’ numbers), and finally to apply them in order to answer ‘contextual’ questions. So,
for example, in the teaching of fractions, formal notions of equivalence though ‘doing the
same to numerator and denominator’ are quickly developed with halves and quarters and
then extended to thirds, fifths, etc. The idea of a common denominator is also introduced
early in the curriculum, and becomes the sole method for comparing and ordering fractions
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and then for addition and subtraction. Even given recent moves towards spending more
time on a topic, and working with the issue of ‘mastery’ (NCETM, 2014), there are at the
time of writing very few examples of willingness to slow down the process of formalisation.
1.2.1 Mathematics classroom cultures: students’ expectations and experiences
English mathematics education traditions have had a well-documented impact on classroom
cultures and on student experiences and expectations. The emphasis on student
performance in public examinations frequently leads to classroom cultures that emphasise
getting right answers over understanding (Noyes, Drake, Wake & Murphy, 2010; Wake &
Burkhardt, 2013). Consequently, many young people see mathematics as a question of
learning rules which lead to answers based on received wisdom and the authority of the
teacher (De Corte, Op ’t Eynde, & Verschaffel, 2002). It is seen as irrelevant to everyday life,
and as meaningless and abstract (Boaler, 2002).
It is widely acknowledged that many students become disaffected with school mathematics
(Swan, 2006; Nardi & Steward, 2003; Lewis 2011). The Smith Report (Smith, 2004) expressed
concern about the negative attitude and disengagement of many students, and it
highlighted in particular the fact that many students found GCSE Mathematics irrelevant
and boring. Disengagement is compounded for Post-16 GCSE resit students by the fact that
they have already experienced failure, known to have a detrimental effect on students in
terms of motivation levels, confidence and attitude (Boaler, Wiliam & Brown, 2000; Dalby
2013; Hannula, 2002). Resit students are also highly likely to have been taught in lower
ability groups, but many studies report that ability grouping has adverse effects on lower
groups (Francis et al, 2017). Higgins et al. (2015) report that low levels of self-confidence are
responsible for their finding that low attaining learners drop behind by one or two months a
year in comparison with similar students in mixed ability classes, particularly in
mathematics. In lower ability groups, they are likely to experience a reduced curriculum,
which limits exposure to mathematics and the grades they can attain in public examinations
at age 16 (Boaler & Wiliam, 2001; Boaler, Wiliam & Brown, 2000; Hallam & Ireson, 2007;
Solomon, 2007), and lower expectations from their teachers (Horn, 2007; Zevenbergen,
2005).
The patterns of classroom interaction that are fostered by a traditional transmissionist
approach to teaching mathematics can lead students to have lower expectations of
themselves as well as of mathematics. Zevenbergen (2005) argues that lower performing
students’ awareness of the restrictions on them in terms of curriculum and pedagogy leads
them to develop a predisposition towards mathematics as negative and to behave in ways
that contribute further to their reduced participation. Addressing this situation can be
challenging, since interventions which aim to change mathematics pedagogy may be
rejected by students who have become used to particular mathematics classroom cultures;
while they might not like them, they are at least predictable situations in which they have
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developed strategies for coping. An approach which asks students to explain their thinking
and make connections, ask questions and generally take more risks instead of simply
‘learning the rules’ needs to take this into account (Brantlinger, 2014; Lubienski, 2007).
1.3 The challenge of GCSE resit
Teachers of Post-16 GCSE mathematics resit face particular pedagogic challenges as they
seek to raise achievement in a potentially disaffected student body under pressure to
succeed in order to pursue further training and career pathways. ‘Success’ is dominated by
a final examination only months away, and inevitably a large proportion of teaching is
focussed on examination practice, favouring transmission teaching and memorisation of
rules and procedures. Teachers feel a tension between covering all the content (but at an
even quicker pace than when their students first learned it) and taking the time to develop
understanding (Swan, 2006).
The aim of this project was to address the multiple challenges of GCSE mathematics resit –
for both teachers and students - by intervening with a different approach. We saw the
potential of a Realistic Mathematics Education approach for tackling gaps in students’
understanding and replacing poorly retained algorithms with more meaningful approaches
that could support students’ engagement and ultimately raise achievement.
1.4 A note on the GCSE context of this project, and the new GCSE specification
This project took place in the context of the GCSE specification which was last examined in
Summer 2016. The examination could be taken at Foundation or Higher Tier levels, enabling
students to achieve grades C to G (Foundation Tier) and A* to D (Higher Tier). GCSE resit
students do not necessarily take Foundation Tier, and some of the students in this project
were entered for the Higher Tier. Although any grade above a U is a pass grade, a grade C -
termed a ‘good pass’ - is required for entry to further training and jobs. The new 9-1 GCSE
specification, first examined in Summer 2017, includes changes which are particularly
pertinent in the context of the issues discussed in this report, namely: coverage of broader
and deeper mathematical content; a focus in Foundation Tier on core mathematical
understanding and skills; a greater focus on problem-solving; and additional requirements
to provide clear mathematical arguments. At the time of writing, indications are that
students who achieve grade 4 will be classed as gaining a ‘standard’ pass, the minimum level
that they must achieve in order not to be required to continue studying mathematics post-
16. Grade 5 will be described as a “strong pass”. This terminology will replace the
description of the grade C as a "good" pass (TES, 2017).
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2 The Realistic Mathematics Education approach
The aim of Realistic Mathematics Education (RME) is to enable students to visualise
mathematical processes by careful use of context and model-building which is always
present and accessible to the student. The RME approach differs from regular teaching in
that it moves more slowly towards formalisation, and does so in such a way that students
can maintain a link back to the original context that they worked with, and how it has been
modelled. In so doing so it aims to enhance students’ understanding of, and facility with,
mathematical processes which have meaning rather than being rote-learned and
subsequently forgotten or mis-remembered. A potential ‘side-effect’ of more meaningful
mathematics is enhanced student engagement and interest.
2.1 The background to Realistic Mathematics Education
Realistic Mathematics Education is a pedagogical theory developed in the Netherlands over
the last 40 years, shown to lead to greater student engagement, increased understanding of
the underlying concepts and improved problem solving skills (Van den Heuvel-Panhuizen &
Drijvers, 2014). It is internationally recognised, and materials based on RME are used in
many countries (De Lange, 1996) and by over 80% of schools in the Netherlands itself, which
is considered one of the highest achieving countries in the world in mathematics according
to TIMSS and PISA comparisons (TIMSS, 1999, 2007, 2010; OECD, 2017). PISA’s most recent
comparisons showed the Netherlands ranking between 10th and 14th among all participating
countries/economies compared to the UK’s 21st-31st place (OECD, 2016), while an
international comparison of numeracy levels amongst 16-18 year olds by OECD showed the
Netherlands in 2nd position and the UK in 17th (Department for Business, Innovation and
Skills, 2013). Based initially on the ideas of Hans Freudenthal, the RME approach is
significantly different to those used in England in a number of respects. Here we focus on
three of these: the use of context, the use of models, and the notion of progressive
formalisation. We demonstrate why these are particularly pertinent to the post-16 sector
and to GCSE resit students.
2.2 Use of context
The use of context in mathematics teaching is not new. Contexts are often used as a means
of providing interesting topic introductions, and then for testing whether or not pupils can
apply their knowledge. In RME, however, context is used not only to apply previously
learned mathematics, but also to construct new mathematics (Fosnot & Dolk, 2002). In this
respect, context is seen as both the starting point and as the source for learning
mathematics (Treffers, 1987), and contexts are carefully chosen to encourage students to
develop strategies and models which are helpful in the mathematizing process. These
contexts need to be experientially real to the students, so that they can engage in
purposeful mathematical activity (Gravemeijer, Van den Heuvel & Streefland, 1990). Post-16
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students bring a significant amount of ‘life-experience’ to the mathematics classroom which
can be drawn on through the use of carefully chosen contexts which connect it to
mathematics; strategies and procedures are more likely to make sense and there is less
need to resort to memorising rules and procedures. In pilot work, students working out the
‘best buy’ in a supermarket already had a number of informal strategies which could be
modelled in a ratio table, ultimately leading to more formal ideas.
2.3 Use of models and two ways of ‘mathematising’
In RME, models are given the role of bridging the gap between informal understanding
connected to ‘reality’ on the one hand, and the understanding of more formal systems on
the other. Although some ‘models’ are instantly recognisable as such (for example, the
empty number line), the meaning also extends to “materials, visual sketches, paradigmatic
situations, schemes, diagrams, and even symbols” (Van Den Heuvel-Panhuizen, 2003, p. 13).
Models and contexts support the process of formalisation while retaining the ‘sense-
making’ element, allowing the formal and informal to ‘stay connected’ in the minds of the
students. Models also allow students to work at differing levels of abstraction, so that those
who have difficulty with more formal notions can still make progress and will still have
strategies for solving problems. Consequently, teachers feel less pressure to replace
students’ informal knowledge with formal procedures. RME identifies two ways in which
students engage with mathematics; at one level, they solve the contextual problem under
consideration (‘horizontal mathematisation’), on the other, they work within the
mathematical structure itself by reorganising, finding shortcuts, and recognising the wider
applicability of their methods: this is called ‘vertical mathematisation’ (Treffers, 1987) and
is further explained and exemplified in section 4.1.
Vertical mathematisation, and particularly the recognition that the same model can be used
to approach a variety of problems, is particularly beneficial for students attempting to cover
a syllabus in a short time, as in the post-16 context. It enables unification of elements of the
curriculum which previously they would have perceived as ‘different’. In the pilot study for
this project, it was noticeable how students began to recognise that the model of the ratio
table could be used to answer questions on fractions, percentages, ‘best buy’ comparisons,
conversions and so on, which in the UK are traditionally taught using a range of algorithmic
methods which rely on memory.
2.4 Multiple strategies and formalisation: redefining ‘progress’
RME’s emphasis on building on informal strategies does not mean that formal methods and
procedures are ignored. Teachers are always aware of the need for students to develop
mathematically, and to become more mathematically efficient and sophisticated over time.
What RME does do, however, is offer a very different story of how students and teachers
work towards this aim. While formal notions are there, they are seen as being ‘on the
horizon’ (Fosnot & Dolk, 2002) or the ‘tip of the iceberg’ (Webb, Boswinkel & Dekker, 2008)
14
as illustrated in Figure 2-1. If teachers are not to teach formal procedures, however, they
must be given an alternative, and materials based on RME provide this.
Figure 2-1 The tip of the iceberg, taken from Webb, Boswinkel and Dekker (2008)
Here, for example, while 3
4 is the formal notation, within the ‘main body’ of the iceberg are
a range of informal representations and pre-formal strategies which students could work on
and develop. These are not only seen as desirable but as essential under RME – it is through
them that students are able to ‘make sense’ of formal mathematics. This is particularly
important in GCSE resit classes, where students arrive with a range of mathematical
experiences and a variety of previously taught methods and procedures. It is clearly
beneficial if a teacher can develop these, rather than imposing new methods - RME gives a
structure within which this can happen.
Rather than seeing progress as a matter of taking away context in order to work on more
formal mathematics, in RME progress is defined through the progressive formalisation of
models (Van den Heuvel-Panhuizen, 2003), and in particular the progression from ‘model of’
to ‘model for’ (Streefland, 1985). Initially, the model is very closely related to the specific
context being considered, but eventually becomes a model which can be applied in
numerous mathematical situations (Van den Heuvel-Panhuizen, 2003).
In terms of fractions, this can be seen in Figure 2-2. A problem about how to share ‘sub-
sandwiches’ is initially represented by a drawing of a sandwich, but eventually becomes
represented by a model for the formal comparison of fractions.
15
Figure 2-2 Progression from ‘model of’ to ‘model for’
2.5 Applying RME design principles for GCSE resit
Designing materials for work with post-16 students poses particular problems, since
students must undergo a formal examination at the end of a course that is in practice eight
months long. Hence, it was crucial that host teachers could see a learning trajectory that
gave students the opportunity to work towards formal content coverage. It was also
important, from a design standpoint, that the process of formalisation would not come at
the expense of understanding. Contexts were chosen which allowed mathematical
representations to emerge which supported the formalisation process, and which were
realisable to older students.
So, for example, a piece of work looking at fractions begins with a survey of eating habits in
a local college, with students being put in the position of a new canteen chef. A section of
the results from the survey is shown in Figure 2-3.
Figure 2-3 Early section of material in the survey bar lesson
Students then produce their own survey, shading fraction bars appropriately. In addition to
the main focus on fractions, this activity also leads to work on pie charts in a typical example
of how, within RME, models can help to unify different elements of the curriculum.
16
Subsequent questions and activities aim to develop more formal fraction strategies while
retaining the context and model. So, for example, as we move towards adding fractions, the
following problem is posed. “At a staff meeting, Jan asked people how often they used the
canteen; 1
3 said every day,
1
4 said three or four times a week, and
1
10 said once or twice a
week. How many people do you think Jan asked?” This is one strategy for introducing the
notion of a common denominator. At another point, when 1
4 and
1
6 emerge from a survey,
the question asked is “Jan seems to think she asked 40 people for their opinion. Is this
possible? If not, suggest some numbers of people she might have asked”. Later in the
lesson, as we move towards addition and subtraction, the question is “How many segments
should you use in your bar if you want to work out 1
3 +
1
8 ?” Finally, students are asked to
use a ‘segmented bar’ to show that 2
5 +
1
2 =
9
10 and addition and subtraction of fractions
follows. The major design aim is that students see the purpose of needing a ‘common
denominator’, and that they come to the notion themselves in order to be able to solve the
problems posed. At no point is the idea of a common denominator ‘taught’ to the students.
A further significant issue at post-16 is the amount of prior knowledge that students have.
In order to take this into account, materials were designed so that models could serve a dual
purpose: supporting the development of new mathematics for some students, while also
strengthening the understanding of those students who had already met formal methods
and were reasonably secure with them. The segmented fraction bar is an example of how
this works in practice, giving a visualisation of the fraction and enabling some students to
develop the notion of a common denominator, while allowing others to make sense of and
gain further insights into mathematics they already know.
The tension for designers is to maintain the guiding principles of RME within a relatively
short, examination-focussed course. The nature of GCSE resit courses creates challenges to
all the main principles of RME: maintaining the use of context throughout comes under
pressure from the need to ensure students’ ability to answer more abstract examination
questions. This challenge is even more in evidence with the principle of ‘progressive
formalisation’: within an RME approach, contexts and models guide students slowly towards
more formal mathematics. The danger for GCSE resit is that students are pushed towards
the formal too quickly, and connection with ‘reality’ may be lost. A more detailed
explanation of how activities were designed in relation to the principles of RME can be
found in Appendix 2.
17
3 The GCSE re-sit project
This study built on the findings of a small pilot study carried out in 2013 (see Appendix 1).
The pilot identified a number of potential benefits of the use of an RME approach with GCSE
Mathematics resit classes, including:
Improvement from pre- to post-test performance;
Willingness to ‘have a go’ at a new problem;
Greater understanding of connections within mathematics;
Improved ability to explain and justify strategies and solutions;
Teachers’ recognition of the wider application of models;
Teacher reports on the positive impact of RME, including on their own mathematics
understanding.
The current study aimed to extend our understanding of the application of RME in the post-
16 context with a more rigorous evaluation of the intervention in terms of student test
performance. Alongside this, we aimed for a more detailed qualitative investigation of the
impact of RME on student engagement with mathematics in terms of their approach to
problem-solving, and their perceptions of mathematics and of their own learning. We also
sought to understand the implications of the RME approach for (1) students’ perceptions of
and participation in classroom practice; and (2) our host teachers’ perceptions of how the
challenges of teaching GCSE resit classes might be met.
3.1 Research questions
The study set out to address the following research questions:
RQ 1: How does an intervention based on RME impact on students’ achievement and
attitudes in Post-16 GCSE resit mathematics?
RQ 2: How does an RME-based approach affect student understanding and engagement
with mathematics in terms of (a) problem-solving strategies (b) experience of and
aspirations in mathematics and (c) classroom participation?
RQ 3: What issues arise in practice in the adoption of an RME approach with Post-16 GCSE
resit classes? How do teachers perceive the role of RME in the context of the challenges of
working with GCSE resit classes?
RQ 4: What are the implications of students’ and teachers’ experiences of an RME-based
approach for its wider application to the mathematics curriculum in this context and
teachers’ professional support needs?
18
3.2 Methodology
We drew on a variety of quantitative and qualitative methods in this project. To address RQ
1, we employed pre- and post-tests (including both mathematical content and attitude
measures) in a quasi-experimental design devised in collaboration with an independent
evaluator (The Centre for Development and Research, Sheffield Hallam University (CDARE)).
CDARE carried out the analysis required for RQ1, as detailed below. CDARE also observed
one teaching session and interviewed host intervention teachers as part of their process
evaluation of the quasi-experimental study. Their interviews with teachers also aimed to
provide an additional independent perspective in relation to RQ 3. We employed a case
study methodology to address RQs 2, 3 and 4 through a mix of qualitative analysis of
students’ answers in the pre- and post-tests, semi-structured interviews with students in
both control and intervention groups, semi-structured interviews with our host teachers
(both control and intervention class teachers), and lesson observation.
3.2.1 The study sites
Three institutions in two cities in the North-West of England hosted the study, providing 4
intervention/control pairs. All were delivering GCSE resit in 9 months. We refer to them as
follows in this report:
SFC: SFC is a sixth form centre which is part of a large multi-site FE College. At the time of
our study, it was running several resit classes for 16-17-year-olds, limited by college policy
to those with a grade D. Both classes were taught by experienced teachers. One class
hosted our intervention (SFCP) and the other acted as a control (SFCC).
SFS: SFS is the 6th form of a large 11-18 comprehensive girls’ school. It ran two resit classes
catering for its Year 12 and 13 students, covering the full range of GCSE grades D – U. Both
classes were taught by experienced teachers, although one was new to resit that year. This
teacher hosted our intervention (SFSP) and the other acted as control (SFSC).
FEL: FEL is a large multi-site FE college. It ran resit classes for school leavers and younger
students, and mixed age range (17-60 years old) classes. Two paired school leaver classes
hosted our intervention (FELLP) and acted as control (FELLC), and two paired mixed age
classes acted as intervention (FELAP) and control (FELAC). Classes were taught by two
teachers new to resit (FELLP and FELLC), and two with resit experience (FELAP and FELAC).
3.2.2 The host teachers
All the host teachers were specialist teachers, with the exception of one, who had an
engineering background and had taught GCSE mathematics for the past 5 or 6 years (SFCC).
We refer to them in this report as in Table 3-1.
19
Pseudonym Class Experience
David SFCC Entered teaching in FE in 2005, not trained as a mathematics
teacher but has an engineering background. First taught English
but has taught GCSE mathematics for the past 5 or 6 years.
Mike SFCP Trained as a mathematics teacher, in his 6th year in the college.
Asad SFSC Trained as a mathematics teacher, has been teaching GCSE resit
for more than 5 years.
Tanviha SFSP Trained as a mathematics teacher, in her first year of teaching
resit, but has taught Yr 11 for some years.
Lucy FELLC Trained as a mathematics teacher, new to the college but has
experience in adult education. She has taught resit for a short
time in a previous job, and taught GCSE some time ago. In the
intervening years has taught A-level standard mathematics to
international students.
Peter FELLP Trained as a mathematics teacher, in his first year of teaching
resit, normally teaches functional skills
Kate FELAC Trained as a mathematics teacher, new to FE, previously worked
for 5 years in a 6th form college including teaching resit.
Carol FELAP Trained as a mathematics teacher, has been teaching resit for six years.
Table 3-1 The host teachers
3.2.3 The students
Seventy-five students participated in the intervention classes and 72 in control classes. The
final student sample is presented in Table 3-2.
Control Classes Student numbers Intervention classes Student numbers
SFCC 13 SFCP 17
SFSC 23 SFSP 22
FELLC 13 FELLP 13
FELAC 23 FELAP 23
Total 72 75
Table 3-2 The students
3.2.4 The intervention
The intervention consisted of two modules covering the Number and Algebra strands of the
curriculum, using approximately 12 hours and 9 hours of teaching time respectively. The full
lesson schedule for each module can be found in Appendix 3, and drew on our experiences
20
in the post-16 pilot described in Appendix 1. See also Appendix 2 for an explanation of the
design principles. The Number module was taught in the Autumn term 2014, and the
Algebra module in the Spring term 2015, as in Table 3-3. Exact timing was negotiated with
host intervention teachers according to their overall plan of work, and also taking into
account control group teachers’ plans.
Phase Pre-Test (beginning of module) Post-test (end of module)
Phase 1: Number, Autumn
2014 – 12 hours
Number Number
Phase2: Algebra, Spring
2015 – 9 hours
Algebra Algebra
April/May 2015 Delayed post-test: Number, algebra and other mathematics
Table 3-3 Phases and testing
Teaching was delivered by three members of the research team, two taking responsibility
for one intervention class each (FELAP and FELLP), and the third taking responsibility for two
classes (SFCP and SFSP). All are experienced mathematics teachers with longstanding RME
experience. They took sole responsibility for planning and teaching in their classes, with host
teachers sometimes present and sometimes not; if they were present, they were
encouraged to observe but not participate in order to avoid the impact of an increased
teacher-student ratio. Intervention teaching followed the research team’s agreed planning
of the modules, but team members made their own decisions on the pace of delivery
according to local conditions. These included timetabling (for example, FEL ran 3-hour
classes once a week whereas SFS and SFC ran 90 minute or 2 hour classes twice weekly), and
differences in student progress, as exemplified in the delivery of the survey bar lesson in
section 4.2.1.
3.2.5 Data collection
We collected a variety of data during and after the intervention. Details of the pre-post
tests, questionnaires and interview topic guides are presented in a Technical Appendix,
available from the authors on request. We obtained informed consent from all students and
teachers at the beginning of the project. All were promised confidentiality and anonymity,
and assured of their right to withdraw at any time. We obtained specific permission from
students for videoing classes, on the assurance that there would no focus on specific
individuals. Pseudonyms are used throughout this report.
3.2.5.1 Pre- and post-test, and the delayed post-test
All project and control students took a short test prior to and at the end of each module,
and a delayed post-test. Test items were devised by the research team with final test design
21
undertaken with advice from the CDARE evaluators, particularly in relation to question
construction which would enable identification of evidence of the use of RME intervention
methods (see Section 4.1 for further details). The tests contained a range of typical GCSE
questions on the target subject matter, which had been trialled during the pilot project and
amended in order to reveal differences in levels of student understanding of the underlying
mathematical concepts. The delayed post-test covered both number and algebra, and a
selection of other GCSE areas. This test was delivered as late as possible after the phase 2
teaching intervention in each site, subject to negotiation with host teachers. The tests were
blinded with respect to site and membership of control versus intervention groups, and
were marked and internally moderated by the Manchester Metropolitan University
researchers. A random selection of papers was sent to CDARE for checking as part of their
external evaluation process, and CDARE carried out the analysis of impact as described in
their report in Appendix 5.
Three students from each intervention class were selected on the basis of their post-test
solutions for both number and algebra and asked to elaborate on their solution strategies in
short video-recorded post-test interviews. These data informed our general understanding
of student engagement with the RME materials.
3.2.5.2 Student attitudes and experience
All project and control students completed an attitude to mathematics questionnaire prior
to the intervention and after the second module delivery. This was adapted from the
Understanding Post-16 Participation in Mathematics and Physics Project (Reiss, 2012), with
additional items designed to identify beliefs about mathematics.
A sub-sample of students (13 intervention students and 14 control students, spread across
the 8 classes) participated in semi-structured interviews during the delivery of the first
module for intervention students and at a comparable point in time for control students.
Discussion focused on prior and current experiences of mathematics learning, perceptions
of mathematics and aspirations to study it in the future. Project students were also asked
about their experience of the intervention modules. In addition, seven intervention students
provided ‘echo smartpen’ data on one or in some cases two occasions. Smart pens record
written work in addition to speech; when using the pens a member of the research team
was present in order to elicit student accounts of their thinking.
3.2.5.3 Teachers’ experiences of GCSE resit
All eight teachers were interviewed during the number module phase. One (Mike, the
intervention teacher at SFC) participated in an additional interview during the algebra
delivery, partly in response to his clear interest in the project but also in response to his
anxiety about time pressure. Discussion focussed on the challenges of teaching GCSE re-sit,
their usual pedagogical approach, and their perceptions of the RME intervention in terms of
22
their views on students’ responses, issues such as pace and coverage, the potential impact
of the intervention on their own practice, their experience of working with the Manchester
Metropolitan University team, and their anticipated future practice and professional needs.
3.2.5.4 Lesson observation and recording
We video-recorded a large majority of intervention lessons, using a fixed camera focused on
the teacher. Most lessons were also observed by one of the research team, who noted
students’ responses to the materials and to the central elements of RME. We were most
interested in the ways in which students reacted to the different socio-mathematical norms
of an RME approach, and on their participation in different forms of interaction and sense-
making in mathematics. We explore this issue in Section 4.2.
3.2.6 Independent evaluation
An independent evaluation of the RME intervention was led by Mark Boylan from the
Centre for Development and Research in Education (CDARE), Sheffield Hallam University,
assisted by Tim Jay of Sheffield Hallam University. The aims of the evaluation were:
(i) To provide an independent evaluation of the impact of the RME approach as
operationalised in the project, on Post-16 GCSE resit students' achievements and attitudes
in mathematics, by contributing to addressing RQ1.
(ii) To advise the Manchester Metropolitan University team on issues of fidelity and teacher
CPD, so contributing to addressing RQ3.
(iii) To advise on scalability of the intervention for a larger efficacy trial using a randomised
controlled trial methodology.
The evaluation report is reproduced in Appendix 5, where details of the pre-/post-test
analysis process and outcomes can be found. We draw on these findings in section 4.1 of
this report, where the main results of CDARE’s analysis are presented prior to analysis of the
qualitative data collected and analysed by the research team. We also draw on CDARE’s
reflections and recommendations in sections 8.5.6-8.5.7 in formulating our own implications
and recommendations in section 6.
23
4 Research findings: the impact of RME
4.1 Overall test performance and script analysis of the Number test
As the independent evaluators, CDARE took responsibility for analysis of the test data, and
their full analysis is presented in detail in Appendix 5. Their analysis of scores from 75
(intervention) and 73 (control) students showed small but significant gains for the
intervention group on the Number module post-test scores (F1,93=4.55, p=0.035, Cohen's d =
0.26). There was no effect on performance in the Algebra module at test result level; we
comment on this finding in Sections 5 and 6.
In Number, the scores for both groups increased and revealed a similar level of performance
in post-tests, but a lower level of performance by the intervention group at the pre-test
stage. Further analysis led the evaluators to suggest that the intervention had been effective
and had resulted in the intervention group ‘catching up’ with the control group.
In addition, there was a significant difference between intervention and control groups in
terms of the use of RME methods to answer questions. Of the 49 students in the
intervention group who took the Number post-test, 36 students were seen to use a bar or a
ratio table at least once. There was also a significant correlation between students’
improvement from pre- to post-test in Number and the extent to which they used an RME
approach (r = .258, n = 86, p = .016).
In order to investigate these findings further, we carried out additional in-depth script
analysis on pre- and post-test performance in Number, as this was where intervention
students made most progress. Our aim was to analyse what strategies students found
useful and make comparisons with the ‘Concepts in Secondary Mathematics and Science’
study (CSMS) from the 1970s. In particular, we examined how the Realistic Mathematics
Education (RME) approach enabled students to make progress. Our analysis involved:
Comparing marks gained from pre-test to post-test for each question;
Categorising and quantifying the types of responses given for each question;
Scrutinising how the use of RME methods impacted on individual student approaches.
(As noted in the external evaluation, missing data was an issue. In this analysis, we are able
to report on 54 intervention students and 39 in the control, ie, students for whom we had
both pre-test and post-test Number scripts.) The basic marks comparison is shown in Table
4.1.
24
Question 1. Photocopier 2. Sharing in given ratio 3. 17% of £3300
Pre Post Pre Post Pre Post
Intervention 42% 47% 27% 56% 40% 54%
Control 51% 45% 44% 55% 43% 54%
Question 4. 𝟓
𝟖 of £600 5. Comparing speeds 6. Reverse percentage
Pre Post Pre Post Pre Post
Intervention 31% 41% 20% 42% 6% 18%
Control 33% 27% 24% 42% 0% 0 %
Table 4-1 Percentage of marks gained per question by intervention and control groups in Number pre- and
post-tests
The Number test was designed to consist of GCSE-type questions covering grades B to E
involving the topics ratio, proportion, finding a percentage of an amount, finding a fraction
of an amount, comparing two rates and a reverse percentage calculation. Quantities were
chosen which students could be expected to manipulate without the use of a calculator.
Two out of the six questions analysed were similar to questions used in the Ratio and
Proportion test taken from the ‘Concepts in Secondary Mathematics and Science’ (CSMS)
1970’s programme which was revisited thirty years later, as part of the ‘Increasing
Confidence and Competence in Algebra and Multiplicative Structures’ (ICCAMS) programme.
A detailed analysis of student performance on these two questions forms the basis of this
section.
4.1.1 Methods used by the students to answer question one
4.1.1.1 Use of formal methods
There are well-known formal approaches to solving problems of this type. One is to use a
formula of the type 𝑎
𝑏 =
𝑐
𝑑 substituting in the three known values and solving to find the
fourth. The second is the unitary method. i.e.
Question 1: It takes a photocopier 18 seconds to produce 12 copies. How long
would it take at the same speed to produce 30 copies?
25
None of the students in this study presented solutions using the formula 𝑎
𝑏 =
𝑐
𝑑 and only
three students attempted the formal unitary method. This finding is consistent with the
analysis of secondary students’ performance on ratio and proportion items carried out by
the CSMS study where use of either of the two formal methods was extremely rare. The
CSMS study commented that one class consistently and correctly used the 𝑎
𝑏 =
𝑐
𝑑 rule but it is
worth noting that these were a group of high attainers. As with the CSMS study there was
little overall evidence that this cohort of GCSE resit students were familiar with the formally
taught methods. They will no doubt have seen these methods in school textbooks or
teacher demonstrations or various revision support resources, but on the evidence of this
study, they have had little impact.
4.1.1.2 Informal versions of the unitary method
Some of the GCSE resit students did attempt solutions which involved working out the value
of one ‘unit’, but these were structured in a less formal way than in the unitary method
shown above. Around 10% of intervention students used informal unitary type methods in
their pre-test and all were successful in this, but interestingly these students all elected to
use a bar or ratio table in their post-test. (When interviewed these students spoke about
how they found it easier to draw a bar or a ratio table because they could do that to answer
lots of questions. It would appear in the case of these students that exposure to an RME
approach has encouraged them to move away from multiplicative reasoning (30 copies in 30
x 1.5 seconds) to building up approaches involving doubling, halving and adding amounts.
This could be seen as a backwards step and yet these students all gained more marks in the
post-test because they were able to see the generalisability of the RME strategies to answer
a range of question types.)
For the control groups, just over 20% of students used informal unitary type methods in pre-
test and a similar amount in the post-test, although this was not always the same students.
The success rate for control students using this method was 8% percent in pre-test rising to
13% in post-test. Most students were able to identify a need to do 18 ÷12 to find the time
for one copy, but either left this blank or re-wrote the division formally as 12 l 18, at which
point they then became stuck. Dividing one whole number by another proved to be a
sticking point for many students throughout the study.
12 copies takes 18 seconds
1 copy takes 18
12 seconds
30 copies takes 18
12 x 30 seconds
26
4.1.1.3 Building up an answer
‘Build up’ methods (Hart 1981; Küchemann 1981) involve doubling, halving, adding
combinations of these, trebling, scaling by 10 or other scale factors. Such strategies were in
frequent use in the CSMS study, with students tailoring their approaches to suit the
numbers given in the question. Likewise, the GCSE resit students favoured a ‘building up’
approach. In the case of the photocopier question, where 12 copies are produced in 18
seconds, the most common version was to double these quantities, halve the quantities and
then to add: 24 copies + 6 copies takes 36 seconds + 9 seconds = 45 seconds.
Of the intervention students, 61% used a building up approach in the pre-test rising to 74%
in the post-test. The corresponding figures for the control are 49% (pre-test) rising to 59%
(post-test). However, in both the intervention and the control groups, not all of these
students were able to build to a correct solution. In some cases, students doubled only, or
doubled and trebled and stopped or, having exhausted the possibility of reaching 30 copies
through use of whole number multipliers, resorted to an additive strategy. The incorrect use
of an additive strategy for quantities which are in a ‘rated’ relationship is well documented
(Hart et al, 1981; Lamon, 1999; Küchemann, Hodgen & Brown, 2011), and of the students
who attempted a ‘build up’ strategy, around one quarter demonstrated this error.
4.1.1.4 Evidence of impact
The external evaluation suggests that the intervention had been effective in enabling the
intervention group to ‘catch up’ with the control. In order to investigate this further we
calculated the percentage of marks achieved by each group in pre- and post-tests on each
question. The decision to look at marks as opposed to numbers of student who were able to
achieve the correct answer reflected the need to account for students who were able to
achieve one of the two marks awarded to this question. The results for the photocopier
question are shown in Table 4-2:
Photocopier question Percentage of marks gained in
pre-test
Percentage of marks gained in
post-test
Intervention group 42% 47%
Control group 51% 45%
Table 4-2 The photocopier question results
Clearly, the gain in percentage marks for the intervention group is only slight, as is the
corresponding drop in marks for the control, suggesting that there had been little significant
impact of either the intervention or the control teaching phases on a student’s ability to
gain marks on this question. However, the trend for the intervention group to increase the
percentage of marks gained from pre- to post-test by more than the control group was
27
replicated in every question (see Appendix 5), particularly when their initial marks were
considerably lower than the control groups, which validates the ‘catching up ‘ effect.
What is significant for this question is how many intervention students shifted to using an
RME type approach in their post-test – 63%. Comparing the control group pre- and post-test
scripts, there was no such shift from one strategy to another and no emergence of a
particular strategy which could be aligned with the way a teacher had taught during the
control phase. With the intervention groups it was clear that several students were
attempting to answer this question using either a bar model or a ratio table model. In the
next section, we look at specific examples of how students applied these models and how
using these approaches helped students to make progress.
4.1.1.5 Examples of RME strategies enabling students to make progress
Student A was a 17 year old resit student who entered Year 12 having gained a grade E in
her GCSE. The school’s policy was to enter most resit students for higher level GCSE in the
belief that it was easier to gain a grade C. In her resit at the end of Year 12, student A gained
a low grade D. Student A was attentive in lessons, but like many of the group lacked
confidence in her ability to do mathematics.
In her pre-test (Figure 4-1), Student A appears to be making a genuine attempt to engage
with the meaning of the statement ‘it takes a photocopier 18 seconds to produce 12 copies’.
At one stage, she draws 12 tally marks. Her use of the words ‘it will take them…’ implies an
attempt to make sense of her doubled values in terms of the context of time and copies.
However, she incorrectly swaps the copies and seconds and a close inspection of her vertical
Figure 4-1 Student A, photocopier question, pre-test attempt
28
arithmetic would suggest that her 36 has come from adding three lots of 12 rather than
doubling 36. In her own words, she is ‘confused’.
In her post-test, Student A draws a ratio table and is now able to use a ‘build up’ strategy.
She realises that 30 copies can be made from the time taken for 24 + 6 copies, but proceeds
to make an error when adding 36 and 9 seconds. The crossings out would imply there is still
a good degree of uncertainty, but some progress afforded by the structure imposed by use
of a ratio table (Figure 4-2).
Student A scored zero marks in her pre-test, where she used a variety of half-remembered
rules and referred to not being able to ‘remember it’ (ratio); don’t know how to do these
types’ (percentage); ‘not sure’ (fractions) and ‘don’t have a clue’ (comparing speeds). In her
post-test she made use of the ratio table as shown in Figure 4-2 and was able to represent
the other questions using a bar model. By accurately portioning her bars, she was able to
figure out 5
8 of £600 and find 10% and 1% of £3300. This only amounted to 4 marks, but we
see this as a good deal of progress in that she has moved away from trying to remember a
separate rule for each question and perhaps more importantly is beginning to work with a
structure from which she can develop her thinking in a way that makes sense to her.
Student A would need further exposure to this methodology, but it provides her and her
teacher with something to work from as opposed to re-visiting what for this student
appeared to amount to meaningless routines.
Student B also entered Year 12 with a grade D at GCSE. She was under the impression that
she had narrowly missed the grade C and believed that she just needed to ‘brush up’ on a
few topics. She lacked motivation in lessons and was initially reluctant to engage with the
RME approaches.
Figure 4-2 Student A photocopier question, post-test attempt
29
In her pre-test attempt at the photocopier question (Figure 4-3), Student B correctly
identifies 18 ÷12 but is unable to accurately compute this, or know how to use this rate to
find the time for 30 copies. In her post-test (Figure 4-4), she initially represents the
information on a bar then proceeds to use a ratio table. Her ‘build up’ strategy is the same
as that of Student A, and she is able to complete this with accuracy and so achieve full
marks. Student B missed out most of the other questions in the pre-test or used a mis-
remembered rule, i.e. for 5
8 of £600 she wrote 600 ÷5 = 120; 120 ×8 = 960. In her post-test,
she used an RME method for four out of the six questions and as a result gained nine marks
from pre- to post-test.
A noticeable issue for many of the GCSE resit students was their lack of ability and
confidence to accurately and consistently operate with two and three digit numbers.
Inappropriate application of vertical algorithms, as seen in Student A’s solutions, plus errors
in mentally adding, subtracting and dividing numbers by 10 were quite common. There were
many other examples where drawing an RME model liberated students in terms of what
Figure 4-4 Student B, photocopier question, post-test attempt
Figure 4-3 Student B, photocopier question, pre-test attempt
30
operations to perform, but a lack of fluency in basic number meant they were not able to
successfully compute the required operations.
The ratio table was the more popular choice of method for the photocopier question with
twice as many students using a ratio table compared to those using a bar. In the next
section we illustrate how use of the ratio table encouraged students to take ownership of
their work and develop individual pathways towards solving the problems.
4.1.1.6 The ratio table
Previous studies (e.g. Middleton & Van den Heuvel-Panhuizen, 1995) have noted the
flexibility of the ratio table as an open computational tool, which can give rise to a range of
student approaches. In the post-test examples in Figures 4-5, 4-6 and 4-7, students use
combinations of multiplicative and additive strategies, use differing numbers of steps and
create larger and smaller entries. In some cases, the quantities are presented in size order,
similar to a scaled bar, in others the entries are entered from left to right.
Figure 4-5 Student C’s ratio table strategy for the photocopier question
Figure 4-6 Student D’s ratio table strategy for the photocopier question
31
The ratio table provides a medium for students to organise their thinking and keep track of
operations and results. It encourages students to be creative; there were no examples of
the novel ‘build up’ routines shown above in scripts where students did not use an RME
method. It also leaves an evidence trail of the thought processes of the students, which is
useful to learners and teachers alike: it is easy to identify that Student C made a calculation
error as opposed to a process issue. In the classroom situation, it provides material for
students to compare and contrast their approaches: Student D did not see the ‘popular’
strategy of combining 24 and 6 copies, but instead went on to find and then double 15
copies. This kind of novel thinking inspires questions and debate. It is not clear whether
Student E combined 6 and 24 copies to make 30 copies, or whether she scaled up from 1
copy to 30, but her ratio table demonstrates that not only are several different routes
available within the course of one solution, but that these lead to the same answer.
One disadvantage of this flexible use of the ratio table is that it may encourage students to
work less efficiently, and increase the number of entries made; this increases their chance
of making errors. As previously mentioned, this is a particular concern for GCSE resit
students where their facility for number operations is often low. However, this issue has to
be balanced against the flexibility it promotes, enabling students to ‘own’ their method.
Once they have set up their ratio table the learner is encouraged to fill in ‘what else do you
know’ before focusing on the requirement of the question. This is empowering and
confidence boosting for GCSE resit students who have struggled for many years to
remember and replicate the precise steps of a particular method owned by their teachers
and their textbooks, not by them.
Middleton and Van den Heuvel-Panhuizen (1995) describe the ratio table as ‘a simple
tool.for developing students’ conceptual understanding of rational number’. In other words,
it enables students to develop a number sense around seeing the connections across
fractions, decimals, percentages and ratios in terms of how their individual notations link
and how they are embedded in a variety of different situations. Certainly, fractions,
decimals, percentages and ratios can all be represented in a ratio table, so it does provide a
model where seemingly different notations and the contexts leading to those notations can
Figure 4-7 Student E’s ratio table strategy for the photocopier question
32
be seen to be equivalent. In terms of this study, it is difficult to judge how much the
intervention helped to develop students’ conceptual understanding of rational number, but
at the very least many post-test students were able to recognise that questions which they
had attempted in pre-test using a variety of different methods and which traditionally would
be classed under different topic headings, could now be answered using the same model.
Student D, who successfully used a ratio table to answer the proportionality question (see
figure 4-6), was also able to apply this approach to compare two rates in question 5. In the
pre-test he had correctly identified the division required to work out the speed of the lion,
but had been unable to proceed further as in Figure 4-8. Figure 4-9 shows his post-test
attempt.
Figure 4-8 Student D’s pre-test attempt to compare two speeds.
Figure 4-9 Student D’s post-test attempt to compare two speeds.
33
It is worth noting that several other intervention students shifted to a ratio table solution
strategy for questions that they had successfully solved in their pre-tests, using other
methods. When interviewed, some of these students clearly recognised the power of the
ratio table and/or number bar for answering questions across a number of topic areas. In a
‘eureka moment’ during the third lesson, one intervention student, recognising the
tremendous potential of the ratio table raised her hand and said almost in disbelief, ‘What, I
can use this for percentage as well? I don’t have to remember all those other concoctions?’
4.1.2 Methods used by the students to answer question two
The latest statutory programmes of study for mathematics (DfE, 2013) indicate that
students should be introduced to the idea of ratio including the a: b notation in year 6 and
then work on dividing quantities in a given ratio a year later in Key Stage 3. The results of
the 1970’s CSMS study for a similar problem to the one shown above (Hart, 1981) gave the
percentage of students obtaining the correct answer as 44% (12-13 year olds), 46% (13-14
year olds) and 57% (14 –15 year olds). By way of comparison, the percentage of GCSE resit
students obtaining the correct answer to the share in a given ratio question is shown in
Table 4-3.
Share in a given ratio question Percentage of students correct
pre-test
Percentage of students correct
post-test
Intervention group 20% 44%
Control group 38% 51%
Table 4-3 Performance in the given ratio question
It is interesting to note that, even after the intervention, the facility of the post-16 students
to answer this problem is still below the facility of the CSMS 14-15 year olds and serves to
remind us how difficult these problems are for lower attaining students, despite revisiting
the topic of ratio over a number of years of schooling. Prior to the intervention, their facility
was lower than that of CCSMS 12-13 year olds and particularly so in the case of the
intervention group. Further script analysis was completed in order to identify the types of
errors.
Question 2: Pat and Julie share £140 in the ratio 2 : 5. How much
money does Julie get?
34
4.1.2.1 Exemplification of errors and omissions
In the pre-test, 31% of control students left the question out, the same figure as for the
intervention group. If they could not do a question, students were asked to give reasons.
Their comments included: ‘forgotten the method’; ‘can’t do ratio’; ‘need more work on this’;
‘never was any good at ratios so always try to avoid, they look too complicated to be done’;
‘not sure what ratio is’; ‘found it hard to learn probably why I don’t remember how to do it’;
‘need to practice ratio, never got one of these correct’. The use of the word ‘ratio’ in the
question seemed to evoke reactions similar to that to the words ‘algebra’ or ‘fractions’ as
known difficult topics for these students which they would prefer not to study; there is an
awareness that there is ‘a method’ and an acknowledgement of the role of memory in
reproducing this rule. The comparable CSMS question did not make use of the word ratio or
the a : b notation, suggesting that language may be part of the barrier. After re-visiting the
topic, the incidence of non-attempt by post-16 students in the post-test reduced to 21% for
the control, but more so for the intervention group at 11%.
The most common error made by the CSMS students was to equally share the total amount
(240 hours) between the number of people involved in the CSMS problem (3 people). Very
few post-16 students equally distributed the £140, the most common errors being made by
students who appeared to be using half-remembered rules. These included 140 ÷2 and 140
÷5 or even 140 ×7, using what looks like a concoction of the numbers given in the question
linked by the mathematical operations usually involved in answering questions of this type.
An attempt at ‘doing something with the numbers’ irrespective of the meaning of those
numbers leads to a rich array of procedural malfunctions.
4.1.2.2 Procedural malfunction
An over-reliance on the use of procedures and a tendency for students to mis-represent
those procedures is well-documented (Foster, 2014; Ofsted, 2008; Plunkett, 1979; Swan,
2006). In this study, we found many other examples of students attempting to apply a
previously taught procedure including: trying to find 17% of £3300 by writing it as 3300
17
×100; working out 5
8 of £600 by replacing
5
8 with 0.58; using long multiplication with two
of the three quantities given in the photocopier question to give an unrealistically high
amount of seconds. Not only does this reveal their inability to remember a formal
procedure, but also how little sense they have of how or why the rule delivers a sensible
answer to the question. As commented by Hart (1981, pg. 73), rather than looking at a
problem and saying ‘what does this mean?’, instead the student thinks ‘what do I do when
that sign appears?’
It was noticeable in previous RME-based projects how many students relied on the use of
formal procedures and how many, particularly in the case of the lower attaining students,
over-generalised or mis-remembered those procedures (Dickinson, Eade, Gough & Hough,
35
2010). The encouraging finding was that where students had experienced an RME
intervention, they gradually shifted to using strategies which made sense to them in terms
of answering the question, with much greater gains from pre- to post- tests (Barmby,
Dickinson, Hough & Searle, 2011). Analysing the GCSE resit post-test scripts revealed several
examples of students engaging in sense making, using RME strategies. This is illustrated by
post-test intervention group solutions to question two.
4.1.2.3 Attempting to make sense of the problem
Student F’s initial attempt (Figure 4-10) may be an example of a mis-remembered
procedure, or instead an attempt to hand out 2 lots of £140 to one person and 5 lots to the
other. In the post-test (Figure 4-11), she draws a bar split into 7 parts, and uses shading to
distinguish the portions. Her bar is labelled as a continuum from 0 to £140. This time she
selects a division procedure which when interviewed she was able to justify because ‘there’s
7 boxes, it’s 140 for the whole thing’.
In his pre-test (Figure 4-12) Student G is able to complete the first two stages in the
standard procedure, but then says he has forgotten how to do ratio.
Figure 4-10 Student F, ratio question, pre-test attempt
Figure 4-11 Student F, ratio question, post-test attempt
36
In his post-test bar (Figure 4-13), he uses the initials P and J to distinguish the parts and is
able to represent the 20 in the context of his picture. The presence of dots within the blocks
would suggest he has touch counted to reach Julie’s total, in addition to writing it vertically
as a repeated addition sum. A concern is the way he has marked 70 as though it is in the
middle of his bar and yet it appears at the end of three of his seven pieces. This was an issue
for a few students who over-generalised the strategy of marking in what they perceived to
be the middle of the bar.
The third example comes from a student who, while scoring one mark in both her pre-test
and post-test attempts, shows a great deal more engagement with the meaning of the
problem in her second attempt (Figures 4-14 and 4-15).
Figure 4-12 Student G, ratio question, pre-test attempt
Figure 4-13 Student G, ratio question, post-test attempt
37
Student H was vocal in lessons about never having been able to do division. In her initial
attempt, she writes 140 ÷7 = 2 but crosses it out. Drawing a bar affords her various other
ways of trying to figure out the worth of one piece, including halving and halving again,
counting up in twos on a bar to make 14 and guess and check in 20’s to make 140.
The examples above show that the transition to using a bar model to answer this question is
not straightforward nor a magic fix. In lots of ways, it serves to expose even more of the
gaps in the students’ understanding However, the intervention students did make more
progress on this question than the control in terms of marks gained (from 27% to 56% for
the intervention compared with 44% to 55% for the control). The fact that 37% of the
intervention students used a bar post-test, 90% of whom gained marks, suggests that there
is value in using this approach.
Figure 4-14 Student H, ratio question, pre-test attempt
Figure 4-15 Student H, ratio question, post-test attempt
38
4.1.3 The bar model
The bar model has huge potential as a device for enabling students to think mathematically.
By representing situations on a bar, students can visualise, make connections, deduce
information and so make progress towards solving a wide variety of problems in Number
and in Algebra. The use of bar modelling has gained momentum in England in the last few
years as a result of looking to learn from the success of high performing jurisdictions, in
particular Singapore (DfE, 2013). In Singapore, students are introduced to the method
gradually as a natural part of working with Number. Use of the ‘Singapore bar’ is thought to
account for why their students perform so well in problem solving (Englard, 2010). It is
important to note that the way students in this study were introduced to the bar model,
through the use of Realistic Mathematics Education, is subtly different to the approaches in
Singapore and in England.
4.1.3.1 Realistic Mathematics Education and the associated bar models
The role of context is a fundamental part of RME. One of the main reasons for choosing a
particular context for students to work with is that when students make drawings to
represent that context, they produce a ‘model of’ the context which the designer knows has
a potential for developing mathematical thinking. As a result of years of experience, the
Dutch curriculum designers are aware of contextual situations which will lead to bar-like
representations. A subway sandwich becomes a rectangular bar when you draw it with the
ends squared off. Sharing that sandwich fairly, marking cuts on the rectangular
representation of the sandwich and labelling the pieces with fractions, leads to a bar model
picture, which the Dutch would call a fraction bar. Other contexts such as shading a
rectangular shaped theatre to represent the percentage of seats filled would lead to a
percentage bar type of bar model. For some contexts, i.e. marking bottle stops on a race
route, it may be more appropriate to draw a line showing distance on one side and bottle
stop position on the other. This is sometimes described as a ‘double number line’ but would
still be classed as a type of bar model, where the bar has been flattened to look like a line. In
RME, students are exposed to many contextual situations which can be represented by a
‘model of’ that particular situation. Students are seen to make progress when they start to
see the similarities in the ‘model of’ situations, enough to be able to generalise the use of
these models and apply them to other problems. When students have met enough of the
specific context-related bar models as described above, then they may be in a position to
draw and use a bar to represent a situation which is not obviously ‘bar like’. For example,
students who drew a bar to represent the photocopier problem are not drawing a bar as a
‘model of’ the photocopier itself, but are drawing a bar as a ‘model for’ solving that
problem. In RME, this is described as vertical mathematisation (Treffers, 1987), where
students can recognise the mathematical sameness of different problems and are able to
choose an appropriate model to solve the problem.
39
Designing the material for the Number module required choosing contexts that would
naturally lead to a bar model drawing. One such problem required students to show how to
cut up a rectangular pizza so that one person had 4 slices and the other person had 5, and
share the cost according to what they eat. Two strategies for representing this problem are
shown below:
In the first drawing (Figure 4-16), some students drew the pizza first and then portioned it
into 9 slices; their strategy for doing this tended to be guess the size of one slice, draw 8
same sized slices and then extend or reduce the last slice. Others started by drawing one
slice, repeatedly adding slices until they got up to a 9 sliced pizza. Others split the pizza into
3 parts by drawing 2 vertical lines and then split each of the 3 slices into 3 pieces by drawing
2 horizontal lines (Figure 4-17). Students then develop their own systems for indicating
which person gets which slices (one of the post-16 students always used his own initials,
rather than the names given in the question) and how much each person should pay. The
actions of drawing and splitting their bars can prompt mathematical thinking around the
processes of division; that is, when students are considering how to distribute the £10.80
pizza cost amongst the people eating the pizza, they may see it as the whole amount split
into 9 (£10.80 ÷9); or as guess a price for one slice and check it builds up to make £10.80; or
split the cost into 3 parts (£10.80 ÷3 = £3.60) and then 3 parts again (£3.60 ÷3 = £1.20). In
this example, the context of the rectangular pizza and apportioning the cost leads to a bar
model which is very close to the context, a bar model picture which is at the ‘model of’
stage. In a later lesson, students were presented with problems similar to question 2, where
they had to share money in a given ratio. One student said ‘it’s like the pizza problem’,
another commented ‘what, is that what it means when I get one of those questions, that’s
all I have to do?’. These students would appear to recognise that the bar which had
represented a pizza in an earlier lesson could now be used as a ‘model for’ solving the
sharing money problem.
Figure 4-16 Student images for cutting a pizza into
9 equal slices: example 1
Figure 4-17 Student images for cutting a pizza into 9
equal slices: example 2
40
By the time students were completing the Number module, the intention was for them to
be able to see that the bar model and the ratio table could be used as a ‘model for’ solving a
variety of different problems. In RME, the transition between the role of models from a
‘model of’ to a ‘model for’ is usually seen as a long term process. The fact that many of the
intervention group were able to solve a range of problems in the post- test by drawing a bar
or a ratio table is encouraging, considering that the intervention was for only 12 hours.
However, we would see a case for a much longer intervention, to enable more students to
see the power and the potential of RME based models for unifying their approaches and for
developing their thinking.
4.1.4 The problems with division
There were many examples in both intervention and control scripts of the difficulties
students encounter with division. In many cases, students were able to identify a division,
re-write the sum using the standard ‘bus stop’ notation but could proceed no further. In
other examples, students opted to find percentages such as 1% of 3300 by working out 3300
÷100, using the standard division method, seemingly unaware of how inappropriate this is
as a method for dividing by 100. Others applied the algorithm to finding 1
8 of £600, when
informal approaches linked to repeated halving may have proved much easier to perform.
Anghileri, Beishuizen & van Putten (2002) refer to this as pupils failing to recognise the
number relationships involved, and instead reaching for a formal procedure as soon as they
know a division operation is required.
The bar model representation was helpful in the respect that it enabled some students to
see relationships between numbers and, hence, re-engage with informal approaches.
Students unable to find 5
8 of £600 in their pre-tests drew a bar model representation, from
which they could then target the value of 4
8,
2
8,
1
8 and
5
8 as illustrated in Figure 4-18.
The last test question required students to find the original price of a car, when the current
price of £6820 was 20% less than the original price. None of the control students gained
Figure 4-18 Using a bar model representation to find 5/8 of £600
41
marks on this question in either the pre- or the post-test. By representing the problem on a
bar, intervention students were able to see a straightforward route to the solution. An
example of this strategy is shown in Figure 4-19.
The bar model gave students access to strategies such as repeated halving as a means of
working out the answer to division problems. Halving is particularly useful when the divisor
is 4 or 8, but also helps reduce the size of the numbers for any even number divisor. There
were examples in lessons where students had drawn a bar representing, say, £180,
segmented into 12 pieces. By marking in the middle (£90 and 6) and the middle again (£45
and 3) they could see that 3 segments needed to be worth £45. Even then, the value of one
segment was not always immediately obvious: some students used a grouping (quotitive)
metaphor for division and began to count how many 3’s they needed to get up to 45. Others
used a sharing (partitive) metaphor, which amounted to give each segment an amount (say
£10), check the total (£30) and adjust until the overall total is reached. Although these
strategies were long-winded and inefficient, they were necessary and exposed just how
little knowledge these students had of basic number relationships.
Equally alarming was the insistence by some students on always filling in the half and the
quarter way point of their bars. Helpful as this was for 8 segments, it became much less so
when their bar was cut into 7 pieces. Student J in Figure 4-20 went through several cycles of
halving and combining chunks to fill in as far as the 1
16 th way point on his bar, but given that
the bar is split into 7 pieces, these calculations served no purpose in finding one part.
Student J was a very interesting case in that he scored zero marks in his pre-test, in which he
put down answers with little or no working out. In his post-test he was able to achieve half
marks, through the consistent application of drawing scaled bars, and yet his knowledge of
Figure 4-19 Using a bar model representation to solve a reverse percentage question
42
number facts and number relationships was very poor. For him, the bar provided a means
by which he could make better sense of what the question was asking, a way of recording
interim calculations as opposed to storing them in his head. In the questions where halving
strategies were beneficial, he was able to make good progress.
Figure 4-20 Misapplication of the halving strategy
The students spoke very negatively about their ability to carry out division calculations.
Several commented how they had never been able to do that ‘bus stop thing’, but when
asked for alternatives they needed to go right back to repeated addition of the divisor in
very small chunks. In interview, one student’s approach to finding 600 ÷8 was to start listing
the 8 times table. He got past 80 before he realised that it might be helpful to count up in
80’s rather than in 8’s. It was apparent that he was not able to use the standard formal
procedure and that the only method he had to fall back on was extremely inefficient and
prone to errors. Anghileri, Beishuizen & van Putten (2002) stress the importance of building
from students’ informal solutions, of enabling them to structure and shorten their informal
methods in order to develop greater efficiency, but not at the expense of sense making or
loss of ownership. The approach to teaching division in The Netherlands involves staying
with the informal for much longer, evolving the method of chunking so that students
develop efficiency and speed by removing large chunks at a time. Chunking in this way can
become extremely quick but with the advantage that students still have a sense of the size
of the numbers involved.
4.1.5 Summary
The script analysis revealed a number of issues associated with learning topics connected to
multiplicative reasoning. Students appeared to be relying heavily on memory of rules which
led to high incidences of non-attempts and a mis-representation of strategies. Several
43
students talked about not being able to remember how to do a particular question, or a
feeling that they had never been able to do it. There was little evidence, particularly in pre-
test scripts, of students attempting to make sense of a problem. At times, mis-application of
procedures led to unrealistic answers, but students appeared oblivious to the unsuitability
of their answers. Their ability to perform basic operations in number is a major barrier to
success and a lack of fluency with times tables is still an obstacle for some. The concept of
division is particularly challenging with many students showing that they cannot connect
with the standard formal algorithm for dividing two numbers and yet they will have met this
rule repeatedly over many years of schooling. The need for these students to work with
different approaches to learning mathematics is very apparent.
The impact of the RME intervention was significant, not only in terms of usage (36 out of 49
intervention students used an RME model at least once in the post-test), but also in terms of
enabling students to make progress. Where students had left blanks or given final answer
only solutions in pre-tests, solution spaces became full as students sought to use their bar
and ratio table models to find a route to the solution. Drawing a bar or a ratio table
prompted the students to think differently about the problems, allowing some to re-engage
with informal, sense making strategies, as well as providing a structure within which to
organise and record their thinking. This was particularly useful to students struggling with
formal division. In addition, the RME models acted as an assessment tool with diagnostic
potential for the teacher and the learner. Using a bar or a ratio table encouraged the
students to be flexible and creative. Once they had drawn the model they were free to fill in
other quantities as they chose, and this placed much less demand on memory and helped to
boost confidence.
In terms of marks gained per question, pre- to post-test, the intervention group fared
better. Many of the intervention students applied an RME method to several post-test
questions, and this is encouraging for two reasons. Firstly, it shows that students were
beginning to recognise the unifying potential of the models as a strategy for answering
questions across a range of topics in number. Secondly, it suggests that students were
beginning to vertically mathematise and see how to apply their context-specific models to a
whole range of problems. Given the shortness of the intervention, this is very positive and
we would suggest that there is scope for even greater and longer-term impact within the
context of a GCSE resit course with longer exposure to a RME approach.
44
4.2 The impact of the RME approach on classroom interaction
The design of each module in the intervention meant that the same basic structure was
followed in all four intervention classes. Within this framework, there were local variations
which can be understood as in part dependent on students’ responses to an essential
element of the RME approach: its departure from a common interaction pattern of closed
questions and one-word answers, and a corresponding re-positioning of students as
participants in sense-making. This requires the establishment of a different set of
‘sociomathematical norms’ - “normative aspects of mathematical discussions that are
specific to students' mathematical activity” (Yackel and Cobb, 1996, p. 458). From the point
of view of many mathematics education researchers (for example Boaler, 2002), this shift is
generally desirable in terms of student engagement and understanding, but in an RME
approach it is closely linked to its design principles and the need to build on visualisation of
contexts and associated moves from ‘model of’ to ‘model for’ at a pace which supports
students’ developing formalisation. This meant that lessons were not necessarily identical
across sites: the tutors responded to students’ developing understandings and adjusted the
pace accordingly. As noted in the independent evaluation (see Appendix 5), this variation
was in keeping with the design principles of RME and represents a relatively high level of
implementation fidelity. We illustrate these points in this section, contrasting two lessons
covering the same material, in the school leavers intervention classes at FEL (FELLP) and at
SFS (SFSP).
4.2.1 The survey bar lesson
This lesson appears at an early point in the Number module (approximately 5 hours into the
12), and involves developing ways to compare, add and subtract fractions using a bar
(segmented strip). It builds on the ideas introduced in the earlier “Sweet Shop” lesson (see
Appendix 2), of using contexts which are ‘bar like’ when drawn. In this lesson, the context is
about displaying the results of various surveys on a segmented bar. As the context
develops, students are asked to compare the survey results of a year 7 class of 30 students
with those of a year 12 class of 20 students. The aim is to choose a means of representing
both groups, thus encouraging students to work on informal ideas associated with the
common denominator through the context of choosing (or imagining) a segmented bar
which would allow both survey results to be visualised and displayed, and enable
meaningful comparisons. The lesson is designed so that students are likely to come up with
conflicting views about how to compare a preference for Indian food given by 3 people out
of 20 versus 3 people out of 30. The lesson proceeds with numerous survey examples which
raise further questions for students about representation and comparison. Towards the end
of this group of lessons, students are set traditional bare fractions questions such as 𝟐
𝟓 +
𝟏
𝟑 =
?, but the aim is that many will still be thinking in the context of the segmented bar, either
45
by drawing one, or by thinking along the lines of what size bar would simultaneously
represent – in this case - a group of 5 people and a group of 3 people.
Standard methods for teaching addition and subtraction of fractions rely heavily on memory
rather than understanding or making sense of what is done through manipulation of the
numbers. The approach taken in this lesson encourages students to develop their own ways
of thinking about how to add fractions based on meaning and understanding. This lesson
also requires learners to interpret and compare data presented in tables and charts and so
exposes them to GCSE-type problems traditionally associated with the handling data aspects
of the curriculum.
In this analysis, we focus on the following aspects:
1. Teacher-student interaction patterns, specifically turn-taking and wait time;
2. Socio-mathematical norms, specifically (a) teachers’ positioning of students and students’
self-positioning with regard to authority and control over sense-making; and (b) actions
directed towards mathematical explanation and argument.
As a background for our analysis, some findings on the nature of mathematics classes and
on standard patterns of teacher-student interaction are helpful. For example, Boaler (2003)
compared traditional classes with ‘reform’ classes – ie classes based on an open-ended
problem-solving approach as opposed to the more traditional approach in which
mathematical methods are demonstrated by the teacher, and then practiced by students.
Her findings are summarised in Table 4-4.
Traditional classes: use of time ‘Reform’ classes: use of time
Teacher talks to the students, usually demonstrating methods
21% Teachers talked to the students in the whole class
16%
Teachers questioned students in a whole class format
15% Teacher questioned students in whole class format
32%
Students practiced methods in their books, working individually
48% Students worked on problems in groups
32%
Average time spent on each mathematics problem
2.5 minutes
Average time spent on each mathematics problem
6.8 minutes
Total teaching time used 84% Total teaching time used 82%
Table 4-4 Teachers' use of time in 'reform' and traditional classes as reported in Boaler (2003)
In addition to these aspects of time use, which are largely to do with teachers’ choices, we
can also look in more detail at these interactions. One major aspect of the standard IRF
exchange is its sequence organisation: the talk involves sequences of turns, not just
individual question-answer turns. Schegloff (2007) describes the most basic unit of
sequence organisation as the adjacency pair, comprising two ordered turns, in which the
second part is paired to the first in the sense that it is expected (eg question-answer). The
standard IRF sequence comprises an adjacency pair followed by a sequence-closing third
46
turn which is evaluative in some sense (Schegloff, 2007, p 118). Pairing also entails
‘preference’ – in an adjacency pair, there will be preferred responses to the first part of the
pair. Speakers also aim to maintain intersubjectivity, or to restore it if it is lost, and ‘repair’
is needed (Schegloff, 2007). Hence ‘trouble’ (Ingram & Elliot, p. 40) is repaired, preferably
by the speaker whose turn it is, and preferably initiated by them rather than another (self-
initiated self-repair versus other-initiated self-repair). Repair by another person (other–
repair) is least desirable. Silence can be interpreted as trouble, and so when wait time
lengthens, both teacher and student feel the need to speak in order to repair it – hence the
pressure on teachers to initiate repair through repeating, rephrasing or nominating another
student when their initial question is not answered. Pressure to repair in this way can lead
to a loss of the dialogic teaching associated with an RME approach.
Turn-taking in classroom interaction tends to follow identifiable rules, including the
teacher’s right to nominate the next speaker, and that speaker’s right/obligation to
respond. While this is no different from standard conversation, if the teacher does not
nominate the next speaker, they have the right to continue – students do not have the right
to take the turn of their own volition. If a student is speaking, the teacher is either
nominated or has the right to self-select to take the next turn. If the teacher does not take
this opportunity, the student can continue. This pattern of turn-taking is illustrated in Figure
4-21.
Figure 4-21 Classroom turn-taking structure (From Ingram & Elliot, 2016)
Wait time is a key issue in classroom learning, since it is built into the standard IRF pattern
(ie the pause between teacher and student turns). A long-standing finding is that teachers
typically leave less than one second between asking a question and repeating, rephrasing or
even answering it themselves (Ingram & Elliot, 2016). Following Ingram and Elliot (2016, pp.
42-3), we note four categories of wait time:
47
Wait time I (i): pause between teacher finishing and student starting to speak
Wait time I (ii): pause following teacher finishing and then taking the next turn
Wait time II(i): pause following student finishing speaking and teacher taking the next
turn
Wait time II(ii): pause following student finishing speaking and then continuing their turn
Extending wait time of all types is generally seen as beneficial in terms of allowing students
more time to think and respond, with at least three seconds being a common target for wait
time I (ii). Extending wait II times increases the likelihood of students explaining and
reasoning, asking questions, speculating and interacting with each other (Rowe, 1986).
Rowe also reported that extending wait times affected teachers’ contributions, notably
increasing the likelihood of questions which encouraged students to elaborate or explain.
We take Ingram and Elliot’s (2016) approach here, in terms of recognising students’ roles in
the joint construction of pauses. As they point out, turn-taking is an integral part of the IRF
pattern, and from this point of view maximises opportunities for pauses in comparison to
ordinary conversation, say, but it also relies on both parties, who can ‘manipulate lengths of
pauses to achieve pedagogical and other social goals’ (p. 38). This is highly relevant to the
use of an RME approach, which introduces what for many students is a novel way of
approaching talk about mathematics, departing from the closed question-one-word answer
form of transmission teaching that many have been used to.
4.2.1.1 Lesson A - FELLP
This lesson took place in a class of 9 students from the usual class of 13. This was their
second lesson using the RME approach and was 90 minutes long. The PowerPoint slides are
illustrated in Appendix 4. In the following analysis, we refer to slide numbers in this
PowerPoint.
The general format of the lesson involved an initial period of whole class teacher-student
questioning around the context of ‘healthy eating’ and the interpretation of a school
canteen survey of students’ food preferences (see SLIDE 1). The topic of fractions is merely
mentioned in passing (“What fraction would corn be then?”) in the 6th minute, generating
40 seconds of discussion linking the answer (an eighth) to the survey bar. The lesson is
supported by the use of 7 survey scenarios which move progressively towards the use of
fractions, but fractions are only introduced in explicit form into the survey context in the
53rd minute (SLIDES 4 and 5). Teacher-student questioning on this topic and discussion
about fractions of a million in response to a student’s volunteered contribution (“She could
have asked a million people and you could still get those fractions”) follows until the end of
60 minutes. The teacher introduces two new surveys (SLIDES 6 and 7) focusing more on
fractions, what they represent and how they can be compared on a bar. Only at 69 minutes
are fractions presented as bare fractions questions, although they are still closely linked to
use of the bar, and the explicit focus of discussion is how the bar can be used to work out 1
3 +
48
1
8 and similar questions (SLIDE 8). At 79 minutes the students are set to work on exam-type
bare fractions calculations (SLIDE 10) until the lesson ends.
Figure 4-22 Sarah's solution
Alongside this pacing of the role of fractions, the lesson involves the introduction of a
complexity which leads to paying attention to common dominator, as explained above. At
21.45, the teacher presents the key question of comparing 3 out of 20 and 3 out of 30,
which generates 1 further minute of discussion which is not wholly conclusive. A further
task working on this issue is presented in the 23rd minute, lasting for 9.5 minutes. In the 34th
minute, the teacher returns to the comparison question and the idea of fractions: “I was just
asking a moment ago about Indian, which is more popular, and you’ve got three on each
one so it’s quite hard to tell which is more popular isn’t it and what we always do is try get it
to a fraction, so what did you say the fraction was?”. The students offer estimates in a 30-
second exchange, and the teacher then moves at 34.02 to discussion of one students’
solution (see Figure 4-22). This is prolonged and finally ends at 44.56, with general student
agreement that Indian food is more popular in one class (3
20) than the other (
3
30), and an
agreed explanation as to why this is the case. The discussion ranges across representation
of different numbers of people on equivalent bars, the concept of fairness, and opinion
polls. At 44.56, the teacher introduces a new survey (the sandwich survey, slide 3) and at
49.16 asks another ‘which is more popular?’ question followed at 51.44 with another
(“where is egg more popular?”). This generates some disagreement, with students initially
saying both groups are the same but then changing their minds to argue against the
teacher’s ‘devil’s advocate’ “Why do you think it’s year 8? It’s not it’s the same look!”. The
lesson moves explicitly towards using the bar for comparing fractions in the final 20
minutes.
49
Table 4-5 summarises the overall use of time, while Table 4-6 tracks topic and activity
through the lesson in numbered sequences. Table 4-2 is similar to Table 4-1 but breaks
“Teacher questions students in whole class format” down into two categories: teacher
questioning in whole class and teacher-led discussion of students’ solutions. It also
distinguishes between students working on tasks such as using the bar to represent the
survey results, and students solving problems such as ‘how much bigger is 1
2 than
2
5?’. It is
notable that only 15% of class time is spent working on conventional fractions problems.
Lesson A: overall use of time
Teacher talks to the students in the whole class 3%
Teacher questions students in whole class format 39%
Students work on tasks 22%
Whole class discussion of task solutions 17%
Students work on problems 15%
Whole class discussion of problem solutions 0.3%
Total teaching time used 96.3%
Table 4-5 Teacher use of time in Lesson A
Time (mins/secs)
Topic Activity Notable features T=Teacher, S(s)=Student(s)
1. 0-1 [slide#1]
Canteen survey
T introduces the canteen scenario
2. 1-6.31 Interpreting what the survey bar represents
T-S Q & A T’s first question (how many people were surveyed?) is pre-empted by S volunteering an observation about the bar. Students all join in, sometimes talking at the same time. T’s emphasis is on asking Ss to explain answers, and responses are sometimes several seconds long; there are few one-word answers.
3. 6.31-16.20
Class survey on favourite fruit
Student task 1 – draw survey results on pre-printed bars
Students volunteer comments on how they have represented the task and there is a lot of discussion with teacher explaining and comparing strategies.
4. 16.20-19.10
Sharing class survey representations
T-led discussion of solutions
Focus on how students have done the task. Student contributions are up to 20 seconds long.
5. 19.13-20.27 [slide#2]
Canteen survey
T introduces the two groups scenario
Some ‘pure context’ talk about food preferences accompanies this introduction.
6. 20.30-22.45
Comparing 2 groups
T-S Q & A Key question at 21.45 “So […] if you compare the year 7 and year 12 class, do
50
those 2 classes like Indian the same? Yes or no?” T funnels more than elsewhere in this sequence eg “Because there’s only 20 in the class and...?” [rising intonation]. The question appears to remain unresolved.
7. 22.45- 32.16
Comparing 2 groups
Student task 2 – draw survey results on pre-printed bars
T moves to colouring bars, one for year 7 and one for year 12. As students work, T prompts re whether their drawings allow comparison between year 7 and year 12.
8. 32.16-44.56
Comparing 2 groups
T-led discussion of solutions
T first discusses one student’s solution which uses one strip per person for both year 7 (30 people) and year 12 (20 people) – “it’s quite hard to tell which is more popular isn’t it and what we always do is try get it to a fraction”. T then invites another student to explain her solution, which uses 2 and 3 strips per person respectively (Figure 4-22). Discussion focuses first on her strategy but then students begin to argue about ‘fairness’. Finally there is agreement that comparison depends on both bars being the same size.
9. 44.56-52.40
[slide#3]
Sandwich survey interpretation
T-S Q & A on how to represent survey data for comparison
Students volunteer the votes analogy used by T in the previous discussion. Later, the ‘where is egg most popular?’ elicits the wrong answer from one student (they are the same) but immediate disagreement from many others (year 8)
10. 52.40-60 [slide#4] [slide#5]
Teacher survey interpretation
T-S Q & A on ‘how many’ questions
Numerous voluntary contributions in this sequence. Extended discussion about fractions of a million and which numbers could be represented in the survey.
11. 60-62 [slide#6]
Background music survey
T-S Q & A on ‘how many’ questions
T is speeding up here, students call out answers, and a break is called.
12. 62-65.18 Break – students get up from seats to ‘exercise’
13. 65.18-68 [slide#7]
Hot drinks survey
T-S Q & A on how to represent survey data for comparison
Slide presents survey data in the form of fractions only
14. 68-75 [slide#8]
Drawing bars to add fractions slide 1
T-S Q & A as T models answers on board
Students discuss how to use the bar to add 1
3
and 1
8 and arrive at 24 strips.
15. 75-78.42 [slide#9]
Drawing bars to add fractions slide 2
Students work on problems
Teacher supports, several students talk out loud while doing the task.
16. 78.42-79.03
Drawing bars to add fractions slide 2
T-led discussion of solutions
A sequence of closed questions and one-word answers as T moves through each question.
17. 79.03-79.20
Bare fractions slide
T introduces bare fractions exam-
T compares these questions to those students are likely to encounter in the exam.
51
[slide #10] type questions “You’ll have methods in the past that you can’t remember but now you can use the bar to help you”.
18. 79.20-79.59
Bare fractions slide
T-S Q & A as T models first answer
T asks what other numbers could be used to solve problems using the bar, and students respond with multiples of 12.
19. 80-90 Bare fractions slide
Students work on problems
Students work silently at first then talk out loud, sometimes to each other, teacher scaffolds and joins discussions.
Table 4-6 Time, topic, activity and notable features across Lesson A
4.2.1.2 Teacher-student interaction
Interaction in this lesson is characterised by the proactive role taken by students in co-
constructing the discussion. In contrast to the standard classroom interaction patterns
described above, they self-select and initiate new on-topic comments 7 times. Looking at
turn-taking and wait times, the teacher finishes a turn and then self-selects as next speaker
(type 1b in Figure 4-21) only once, in sequence 4 (Table 4-6), following a Time I (ii) wait time
of 4 seconds. Turns are otherwise of type 1a and 2a, with Time 1(i) wait times of less than a
second with 4 exceptions. The first entails a 4-second wait in sequence 2 which appears to
be an invitation to self-repair a wrong answer:
T: How many people do you think there are in carrot and corn?
S: 100
[T checks others agree]
T: And you said 50 for corn…
S: Yeah. No 25.
T: What FRACTION would corn be then?
S: Quarter.
T: Quarter.
[4 secs Time I(i) wait]
S1: Quarter.
S2: It would be an eighth.
S1: Yeah.
T: An eighth? [looking unsure]
S3: Yeah I’m going with an eighth.
T: How do you get an eighth?
S3: it’s 200 and 25 into a 100 is 4.
T: OK … so if these are all 25s.. [turns round to look at group] then it would be an
eighth because you’d have 8 of everything in the whole thing.
The second and third entail 4- and 5- second waits for what turn out to be correct answers
in sequence 4 and sequence 8.
52
The fourth involves a wait time of 5 seconds in sequence 9:
T: Without doing any calculations at all – egg – where is it more popular?
[5 second wait]
S: It’s not it’s the same.
T: It’s the same?
S: Yeah.
S: Yeah it is yeah.
T: It’s 6 in both, 6 people in both.
Ss: Oh no.
S: No it’s year 8 again.
T: You tell me why.
Ss: [All talking at once] 8 times 6 is 40.
There is, then, little evidence of ‘trouble’ in the lesson, in Ingram and Elliot’s sense of
silences which must be somehow filled or responses which are not preferred and require
repair. Students’ contributions are often long and spontaneous, reflecting non-standard
socio-mathematical norms, as illustrated in the next section.
4.2.1.3 Socio-mathematical norms
Remembering that the standard pattern of interaction places the teacher as an authority
and students as passive receivers of knowledge, there are numerous points in this lesson
which indicate that the class is not following traditional socio-mathematical norms in terms
of the teacher’s and the students’ positioning as participants in sense-making. In addition to
the instances when students initiate new topics, or explain their solutions, there is an
episode of disagreement and discussion in Sequence 8, in which students join with the
teacher in establishing what is mathematically correct:
[This sequence picks up the unresolved question from sequence 6 – “…if you compare
the year 7 and year 12 class, do those 2 classes like Indian the same? Yes or no?”. T
has been talking through one student’s (S4’s) solution (Fig. 4-22). The sequence has
been edited to focus on the discussion.]
T: So the top bar is for year 7 and the lower bar is for year 12. How many bars has
she coloured in for year 7? Altogether.
Ss: 60
T: And year 12.
Ss: 60
T: So 60 for year 12 and 60 for year 7. Now erm for Indian, it’s that bit for year 12
isn’t it and that bit for year 7, that’s right isn’t it. … Erm so Indian goes up to there
53
and Indian goes up to there on the second one. Does that tell you much about which
one is more popular?
S1: No cos she’s used two bars per person on the first one and three on the second
one so it’s the same amount of people, she’s just used more lines or bars on that one
so it’s not – effectively it’s three people and three people so it’s the same.
T: OK. What d’you think S3?
S3: [clarifies which strip represents Indian on each bar].
T: Yeah, that’s the Indian for year 12. And that’s the Indian for year 7.
S2: It’s the year 12 cos they prefer it more…
T: What do you think about – S1’s argument was that she tripled the year 12 and
doubled that [year7] and that’s why it’s bigger. Is it fair to do that?
S1: Sir it has to be the same it has to be fair.
T: What do you think S4? Do you think it’s fair to do what you’ve done?
S4: No I just thought it was easier for me.
T: Yeah, on the bar.
S1: If you’d done [inaudible] give the year 7s 60 and the year 12s 40 it would balance
out like.
T: OK. But then we’d have 2 bars of different length that would take us back to the
same situation of hard to compare them.
Ss: Yeah.
T: I like S4’s method cos she’s got the same length. Are we convinced that – well
we’re not convinced that it’s fair.
Ss: [inaudible]
[T works through the columns for year 7 and 12 multiplying entries by 2 and 3
respectively]
T: OK and what do those numbers [year 12 column] come to?
Ss: 60
T: And what do these [year 7] numbers come to?
Ss: 60
T: 60 OK. So if you had 2 year 7 classes of 30 with the same opinions this is what
you’d get.
Ss: Yes.
T: And if you had?
S: Two year 12 classes.
T: How many?
S4: [correcting] 3 year 12 classes .
T: 3 year 12 classes with the same opinions that’s what you’d get.
Ss: Yes.
T: So is that a fair way of doing it?
Ss: Yes.
54
S: Everybody’s got a different opinion so there’s more [inaudible] to do it… the
answers are going to be the same cos it all depends on what people…
T: It’s almost like what we do – you know in general elections they do opinion polls
and they ask people. When they do a poll how many people do you think they ask?
S: Thousands.
T: They do ask thousands. yeah erm. How many thousand do you reckon?
S: Ten thousand.
T: But do you know how many people can vote?
S: Anyone over the age of 18.
T: How many might that be in the UK?
S: 30, 40 million.
T: So they don’t ask 30 or 40 million people, they ask ten thousand people. And then
they generalise on that. And that’s kind of what you’ve done here, you’ve asked 20
people here and then you’ve times’d by three and you’ve got 30 people here and
you’ve times’d by two so we’ve now got something we can compare. […]
Ss: [inaudible]
T: So let’s go back to our question about Indian. So you have 3 people from each
class.
S1: It’s more popular in the year 12 class now. Cos if you’re giving like … one
person’s got 3 votes, if you times that by 3 then effectively 9 people or 9 votes have
gone towards Indian. But whereas for the year 7 you’ve only given them 2 votes so
effectively they’ve got 6 votes.
T: And this is another copy of S4’s, er and this grey bit is the Indian bit, yeah, so for
Italian, which age group prefer Italian?
S4: Year 7?
S1: It’s…no year 12s, year 12s
T: Yep
[…]
T: And how come you can do it so quickly?
S: It’s just like we’re looking at the general size of how big the bar is.
T: Yeah. Can you do it quickly on S1’s?
S: Not as quickly.
T: Because?
[…]
S: The bar’s not the same size.
T: The bar’s not the same size. The bars help us compare. Brilliant.
This type of discussion is both required and generated by the RME materials. The absence
of algorithms encourages students to interpret the contexts they are presented with and
55
engage with their mathematisation. It is significant that bare fractions calculations are not
presented until the very end, and in keeping with the underlying principle of RME, students
are reminded that they can link this back to visualisable contexts.
4.2.1.4 Lesson B - SFSP
Lesson A illustrates how students can respond positively to the RME approach in terms of
high levels of engagement, enabling a smooth progression towards meaningful
manipulation of fractions. Lesson B illustrates some of the challenges presented by GCSE re-
sit students, and how these can be accommodated within the RME approach. In this lesson
we can point to instances where invitations to participate in ways which differ radically from
students’ past experience are readily accepted but there are also passages where the joint
sense-making that RME makes possible required greater effort on the part of the teacher to
generate.
This lesson took place in a class of 19 students. This was their 3rd lesson using the RME
approach and was 103 minutes long. The same PowerPoint was used as for Lesson A.
As in Lesson A, the general format begins with the school canteen survey scenario (SLIDE 1)
and progresses through to the teacher use of the canteen survey (SLIDE 5), following a
pattern of discussion about the slide contexts, followed by work on how to represent them
and discussion of solutions. The topic of fractions is raised by a student in the 4th minute,
and this is picked up by the teacher, but both references are in the context of the difficulty
of explaining why it is difficult to estimate numbers of responses in the SLIDE 1 data. The
teacher returns to fractions in the 56th minute with reference to the “is Indian more popular
in Year 7 or Year 12” question and ‘comparing chunks’. The first explicit reference to
fractions appears in 86th minute, during the Q&A session accompanying SLIDE 4: the data
are presented as fractions and the teacher demonstrates 1
5 of 20 using the bar. Fractions
become even more of a focus in the final 10 minutes of the lesson, where the teacher
spends time on how to represent fractions on a bar so that they will be comparable. The
pace of the lesson is such that the later slides which focus on bare fractions are held over to
the next lesson.
The overall pace is slower than in Lesson A, and we explore this with respect to teacher-
student interaction and socio-mathematical norms below. The complexity introduced by the
canteen survey (SLIDE 2) begins in the 27th minute (just 6 minutes later than in Lesson A in
fact) with a question inviting a comparison between 15
30 and
10
20. This generates over a minute
of discussion, followed by the Indian dinners question in the 29th minute, leading to a
further 2 minutes of disagreement and discussion. Disagreement arises again in discussion
of students’ representations of the data on SLIDE 2, beginning in the 43rd minute and
extending for the next 20 minutes. The next 2 slides take up the rest of the lesson (40
minutes); whereas in Lesson A the teacher moves through these with Q&A sessions totalling
56
10 minutes, in Lesson B the teacher sets the students to work on tasks for both slides,
adding 15.5 minutes of discussion and 17.5 minutes of student work on representing the
survey data. These different teacher decisions appear to reflect the large number of student
voluntary contributions in Lesson A, in comparison to student difficulties and slow
responses in Lesson B.
Table 4-7 summarises the overall use of time, while Table 4-8 tracks topic and activity
through the lesson in numbered sequences.
Lesson B: overall use of time
Teacher talks to the students in the whole class 5%
Teacher questions students in whole class format 17%
Students work on tasks 41%
Whole class discussion of task solutions 37%
Students work on problems 0%
Whole class discussion of problem solutions 0%
Total teaching time used 100%
Table 4-7 Teacher use of time in Lesson B
Time
(mins/secs)
Topic Activity Notable features T=Teacher, S(s)=Student(s)
1. 0
[slide#1]
Canteen survey
T introduces the
canteen
scenario
2. 1-8.05 Interpreting
what the
survey bar
represents
T-S Q & A This sequence is marked by a prolonged section
with long wait times, and students appear
reluctant to hazard a guess on how many people
chose melon; they are unable to explain why
they are finding this difficult. Eventually there is
a breakthrough at 2.40 when one student notes
that not all the sections are equal but wait times
are still long. A key contribution comes at 4.50
when another student returns to the issue of
inequality, prompting T to draw over the slide at
5.45 to show what it would look like if all votes
were equal. Responses become spontaneous
and immediate.
3. 8.05-
21.00
Class survey on
favourite fruit
Student task 1 –
draw survey
T stresses that there is no right or wrong way to
display the information on the class’ fruit
57
results on pre-
printed bars
preferences and that students’ solutions are
likely to differ.
4. 21-
24.15
Sharing class
survey
representations
T-led discussion
of solutions
T focuses on how students have used the pre-
printed bar, particularly those using more than
one block per person and how they have
calculated in order to use the whole bar.
5. 24.15—
24.45
[slide#2]
Canteen survey
T introduces the
two groups
scenario
6. 24.45-
30.38
Comparing 2
groups
T-S Q & A T asks a year 7/year 12 comparison question
(“out of the year 7 and year 12 who preferred
American more”) (15
30 versus
10
20), at 26.37 with a
wait time of 4 seconds before nominating S to
give an opinion. S gives year 7 as the answer
leading to several Ss responding with argument,
including that preferences are equal. T repeats
the various arguments without evaluation and
then moves to the Indian dinners preference
comparison of 3
20 versus
3
30 (28.30). Many
students contribute and there is disagreement
again. T repeats the arguments, finally arriving
at “you’re saying there’s less people so having 3
out of less people you think is more. We’ll come
back to that.”
7. 30.38-
42.59
Comparing 2
groups
Student task 2 –
draw survey
results on pre-
printed bars
T tells students to use 2 bars to compare the 2
groups. T circulates and encourages Ss to think
about the question and not to guess.
8. 42.59-
62
Comparing 2
groups
T-led discussion
of solutions
T focuses on one S’s solution strategy: “So how
did she make her bars come to the same length?
Even though there’s not the same number of
people. Cos that’s not what most of you have
done.” T moves on to repeat the comparison
question (51.56) while demonstrating on S’s
solution that the answer is now clear visually “I
think some people gave an argument to say if
you’ve got three out of twenty that’s a bigger
chunk than three out of thirty”. Some Ss argue
voluntarily that ‘it’s exactly the same’ (53.55)
58
T remarks “So we’re still hearing the argument
both ways and it’s not been resolved for some
people – go on” (54.20). In response to a
further voluntary contribution T offers an
argument concerning fractions and percentages:
“So if I was to do fractions to me one out of
more is a smaller thing than them [pointing to
board drawing]. I know what you’re saying they
look the same but when you’re comparing
things it’s like when you do percentage you have
to put them out of a 100 you have to put them
out of the same amount and here they aren’t
out of the same amount. So fractions-wise I
think this is a smaller chunk of this”. (55.20)
9. 62-65
[slide#3]
Sandwich
survey
T introduces the
sandwich topic
T uses this space as a rest in addition to topic
introduction
10. 65-
75.00
Comparing 2
groups
Student task 3 –
draw survey
results on pre-
printed bars
T sets task: “If you wanted to draw bars to
compare these and you wanted to do what S did
and you wanted to use the same length bar how
many strips would you give to each person?” S’s
talk to each other, T does some explaining at the
board to individuals, and circulates around class.
11. 75.00-
85.09
Comparing 2
groups
T-led discussion
of task 3
T asks 2 students to put their solutions on the
board and asks Ss to explain what they think the
thinking behind the solutions is. Students are
slow to answer but at 81.10 one makes a
lengthy contribution about how to represent
data on bar strips.
T focuses on how to use the bar to represent
comparisons using different numbers.
12. 85.09-
89.58
[slide#4]
Teacher
sandwich
survey
interpretation
T-S Q & A on
‘how many’
questions
This is the first explicit use of fractions in the
lesson. T demonstrates 1
5 of 20 using the bar and
works through the question “How many said
tuna OR egg“.
13. 89.58-
90.23
[slide #5]
Teacher
canteen use
survey
T introduces
59
14. 90.23-
97.55
Drawing bars to
add fractions
Student task 4 T: “how many people do you think Jan would
have asked?” – asks students to think alone and
then on tables.
T prompts: “Think of a number that you can
have a third of a quarter of and a tenth of”.
15. 97.55-
103.32
How to
represent the
three fractions
T-led discussion
of task 4
T spends this time on students’ difficulties with
Task 4. “If you had to show a third [on the strip
diagram], on one of these, how many, how long
could you make the bar and be able to show me
a third?“ Students offer that 200 would not
work, that three strips would work, T asks for
more numbers and asks “could you make it 5
long?”. One student says yes but others offer 6,
8 (rejected by T), 9, 12, 15. An offer of 10 is
rejected by other students. S’s join in generating
multiples of 3 up to 1500.
Table 4-8 Time, topic, activity and notable features across Lesson B
4.2.1.5 Teacher-student interaction
Students are less proactive overall in this lesson, although there are passages where they
are quite animated, particularly towards the end of Sequence 8, where they are eager to
make contributions to the discussion about 3
20 versus
3
30. A striking feature of the lesson is
the length of both Time 1(i) and Time 1(ii) wait times. Time 1(ii) wait times are particularly
evident early in the lesson, in Sequence 1, when the teacher is inviting students to suggest
how many people preferred the fruits. The teacher self-selects (type 1b turns in Figure 4-21)
on 5 occasions, with wait times of 12, 11, 10, 4 and 4 seconds. These all come before a key
contribution at 4.50. There are four wait 1(i) times in this sequence, of 4 seconds (eliciting a
key response at 2.40 about inequality), 7 seconds, 14 seconds and 2 seconds. After the last
of these, the teacher appropriates the student’s suggestion and the interaction continues
with 1a/2a turn-taking to the end of the sequence.
T: And how many people do you think preferred melon, that was their favourite fruit?
[12 second wait 1(ii)] Melon. S1, what do you think for melon [11 second wait 1(ii)]
no idea? Ok what’s stopping you having a guess [10 second wait 1(ii] S2 any idea,
melon – no? Can you see where melon is on the chart and what did you tell us S3?
S3: There were 200 people.
T: Yeah there were 200 people and this is zero and this is 200 so I’m almost asking
how many people were in this bit. So you’ve got the 200 people lined up there and
I’m asking how many people were in this bit. How many people gave that vote. Have
you got an idea S5? [4 second wait 1(i)]
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S5: If you split, if you try and split there’s 200 in one section so there are 4 sections
but they’re not all equal so you could do… oh you mean the top one…
T: Yeah cos I asked about melon …
S5: It’s the smallest one out of all of them so you could guess that it’s - 10? Or.. I
don’t know.
T: Well I think that’s helped what she said, hasn’t it. So it’s split into 4 but the 4
aren’t equal. And presumably when you add all these four up what should you get?
S: 200
T: Yeah. 200 cos there’s 200 people altogether. S6, any ideas? [ 4 second wait 1(ii)]
could anybody tell me about any of the other strips what they think how many
people they represent. Just raise your hand [4 second wait 1(ii)] could anybody tell
me anything about this? You said about splitting it up, S5 [7 second wait 1(i)]
S5: Yes. I thought that … but I’m not sure.
T: You think what?
S5: I think ... fractions … but I’m not sure.
T: You have fractions coming in between did you say? What d’you mean by that?
S5: Erm… it’s hard to explain.
T: Try, try, because somebody will help you I’m sure [14 second wait 1(i)]
S5: Erm [laughs; other Ss laugh and some talk off topic]
T: S6 any ideas?
S6: No.
T: What’s stopping you having ideas then?
S6: Because a lot of different sizes.
T: Right OK. If they were all the same size how many do you think would be in each
one?
S6: 50 each.
T: 50. Yeah because. OK and if they were all the same size, how would they be
blocked up, where would the lines be? Do you know? [2 second wait 1(i)]
S6: [Shakes head]
T: Can anybody help with that question? What S6 is saying is if these were all the
same size there’d be 50 in each, cos 50 and 50 is a hundred and another 50 is 150
yeah.
S: So that gives you 4 sections.
T: Yeah there’d be 4 sections and they’d be equal. So – and they’re not equal – which
was your thing wasn’t it [to other S] so you’d have to make an adjustment on that.
[S6 is nodding all through this]
T: Right do you think this one’s going to be more than 50 people or less.
S: Less.
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T: Yeah because it’s smaller than – can I draw on this? [asking about whiteboard, T
draws on slide] Right if it was 4 equal sections, where would the lines be? That was
the question I asked.
S: You’d have a point in the middle.
T: Right in the middle yeah because that would be 2 equal sections.
S: And one on each side in the middle.
T: And one on each side in the middle. Right and if the whole thing was 200, as I’ve
said there’d be 50 in each. So it’s almost like if I was to fit this one in here.
Ss: It’d be 25.
T: Ah so it’s coming close to 25. Ok. That suddenly made – yeah, you fit this in here,
it looks like it’s got half, that’s 25 people. Ok erm tell me about oranges. How many
people do you think said oranges?
S: Think it’d be 50.
T: S7 what do you think, where do you think they got 50 from?
S7: Cos this is like, the thing that you’ve drawn is like the same size.
T: Because the orange looks very similar to …. Ok .. good.
Elsewhere in the lesson, wait 1(ii) times occur 5 times, with 4, 7, 2, 2, 4 second waits. Three
of these occur in the last 30 minutes of the lesson, and the first 2 in Sequences 6 and 7,
where the Slide 2 dinner choice survey and its representation are debated. Two further
wait 1(i) times of 2 and 6 seconds occur in the last 30 minutes, and another of 10 seconds in
Sequence 7.
4.2.1.6 Socio-mathematical norms
The patterning of Sequence 1 appears to indicate that the students are initially reluctant to
take responsibility for meaning-making, but that this may largely be due to an assumption
on their part that an exact one-word answer is required by the teacher – which of course is
not possible with the information they are given on the slide. Once this is clear, they appear
more willing to speculate. Sequences 6, 7 and 8 emerge as key points in the lesson where
the students engage in discussion. This extract from Sequence 6 illustrates the argument:
T: So out of the year 7 and year 12 who preferred American more [4 second wait 1(ii)]
do you see what I’m asking S8 there, have you got an opinion.
S8: The year 7s.
T: The year 7 preferred it more than the year 12s, go on tell us why.
S8: [inaudible]
T: So initially you’re saying to me you thought the year 7s presumably because 15 is
more than 10.
[Ss all talk at once]
It’s equal miss.
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No it’s not.
Because…
T: OK let’s hear a few arguments about this. S9 what were you going to say?
S9: [lengthy contribution, inaudible]
T: So your argument is along the lines of if you do 30 and you take off the 15 you’re
left with 15 and if you do the 20 students and you take off the 10 you’re left with 10
so what’s that saying.
[Ss all talk at once]
S: That’s not equal
T: Go on S10 what were you going to say?
S10: Miss I think it’s … because the total of the students so it’s going to be a half of
them that chose it.
T: So you’re saying it’s the same because half of them in year 7 said American and
half in year 12 said American
S10: Yeah
T: OK. Ermm- Was Indian more popular with the year 7s or with the year 12
[Ss all answer at once, some say year 12 some say year 7]
T: OK so some people are offering year 12, let’s have some opinions over here. Erm
S11 do you think it was the year 7 or year 12 where Indian was more popular?
S11: Year 12.
T: Because?
S: Because it’s out of 20 whereas year 7 is out of 30.
Multiple Ss talking, one voice heard clearly: That doesn’t make sense.
T: Let’s hear another argument then, somebody said it doesn’t make sense you must
be thinking of something else. S12 what are you thinking?
S12: Out of 30 is more than out of 20 because there’s more students.
T: So you’re arguing it the other way round, you’re saying there’s more students so 3
out of more students is more than 3 out of… whereas you’re saying it’s out of less
students.
S: Miss they’re the same I think.
T: Do you I don’t think that’s what you said.
S: That’s what I meant though.
T: Go on then.
S: I think it year 12 because there are less than in year 7 so basically more people
want it in year 12 than in year 7.
T: So you’re saying there’s less people so having 3 out of less people you think is
more.
S: Yes compared to those…
T: Well we’ll come back to that.
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Although progress is slower in comparison with Lesson A, the students in this lesson are
eventually willing to participate in argument. One reason for this might be due to the
extended wait times earlier in the lesson, which establish these norms.
4.2.2 Summary
The close analysis of these two lessons illustrates a number of features of the RME
approach, including the use of context, the shift from ‘model of’ to ‘model for’, and a slow
move to formalisation maintaining a continued connection to context. It also illustrates
some of the challenges of introducing the RME approach to the resit context, in terms of
students’ responses to the need to engage in ways which are new to them, as in Lesson B –
that the difference for SFSP students was quite substantial is indicated in the student and
teacher interviews in sections 4.3 and 4.4. However, the analysis here also illustrates how it
is possible to maintain adherence to RME design principles through the use of extended
wait times and an adjustment of the overall pace in order to support students’ developing
mathematisation of the context. We return to the implications of this in Section 6.
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4.3 The student experience: GCSE resit classes and RME
GCSE resit students are a vulnerable group in terms of the impact of their prior experience
of learning mathematics on their confidence and motivation. CDARE’s analysis of the
attitude test and ‘before’ and ‘after’ differences did not show any change (see Appendix 5).
However, as the literature review in Section 1 and the analysis in Section 4.2 suggest,
negative perceptions of mathematics are likely to be quite entrenched in this group of
students. This makes it all the more important to understand students’ experience of resit
classes in general, and their perceptions of mathematics. Given the impact of their prior
experience and the nature of mathematics teaching, we were also interested to hear what
intervention students thought about the experience. Hence in this section we focus on the
interview data. We interviewed students in both control and intervention classes to find
out about their experience of failing GCSE and doing resit classes, their perceptions of
mathematics, and – for the intervention students – their experience of RME classes.
4.3.1 GCSE resit stories – why are they here?
The students explained their previous failure to gain a grade C with reference to external
factors relating to schooling but also to qualities in themselves. Some had made a number
of attempts.
4.3.1.1 Teaching at school
School factors included multiple teachers or poor teaching, mis-match between teaching
and exam content, and the impact of setting. They were often associated not just with
failing but with causing students who had liked mathematics in the past to dislike it. Harriet
and Lucy (FELLC) criticise teachers for over-reliance on text books and failing to explain:
… the teacher I had in Year 10 was good, but then he left, so we got another teacher
in Year 11 and she just like didn’t really teach us. She put like a text book on our
thing and we kind of had to learn through that. [What makes a good teacher?]
Someone who’ll explain things properly in a class and not just like throw a text book.
(Harriet)
… if we like asked her to explain it like a certain amount of times she just couldn’t do
it. She’d only explain it like two times. … That’s why I didn’t like it so ... (Lucy)
Andrew (SFCC) had a similar complaint:
… I don’t get shown how to do the questions and answers properly and I forget them
easy and then they never come back to them and then when it comes to the exam I’ll
just forget them and don’t know them…. Like ratio I get shown once and when I ask
for help there’s not much help.
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Sienna (SFCP) described the reverse situation, explaining that, having done badly at primary
school, she had liked mathematics since entering Year 11, because her teacher did not rely
on a textbook:
I never used to like it until I got into Year 11 because I got like a teacher that I
actually understood, because I used to have teachers who didn’t really teach you
anything, like you never learnt. … He just, he actually wanted to help and like taught
you properly how to do it and went through it if you need help and the other teachers
just give you a text book.
Levi (SFCP) explains that he was a top set student and doing well from years 7 to 10.
Mathematics had always been his favourite subject but he became bored of it towards the
end of year 10 because it was too repetitive in the run-up to GCSE:
It was just like because from Year 7 to Year 10 no one really put pressure on us. …
Like we didn’t have to like stay behind after school. … but then in Year 11 we had to
do it all the time and that like, that was like every day and every lunchtimes and
every after school and it just got boring doing the same thing over and over again
that I weren’t really into doing.
He blames his failure to gain a C on the fact that teaching had focused on certain areas of
the syllabus only:
… they weren’t teaching us what we needed to know, because like in the test half of
the things that was in it I’ve never seen before. Like I felt like they should be teaching
me my weaknesses not my strengths. … but there was questions that … I would skip,
but I couldn’t do that in the test because I had to answer all the questions, so I have a
chance of getting more marks, so I prefer like the teacher should’ve taught me things
that I wasn’t good at. Instead I was given the work that I was good at, that I would
just finish like that.
Ruqya (SFSC) told a similar story, demonstrating how aware of marks the students are:
… the questions they weren’t like what I expected to see in the paper. I don’t think
the lessons prepared me that much, cos we used to like to focus more on the
questions that were like 5 marks. But those were the sort of things that I didn’t really
need help on. … But I was like quite good at them, it was more like questions got to
do with like equations and percentages. [And how many marks go with those
questions?] For equations there’s about 10 marks.
4.3.1.2 Setting
The 6th form students raised a different issue: the demotivating effects of setting, and its
related teaching. Imogen (SFSP) was in ‘a pretty bad set. I was in set 8’:
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I’ve got a long experience of people telling me that I’m not good at maths. … Like
even my parents say it. They’re like you’re not that good at maths. It’s not really
your strong point. … So you kind of, you kind of like when you get told something
enough times you’re kind of like “okay I’m not good at maths”. … I didn’t really like it
[in set 8] no and I think that’s partly the reason why I failed as well, because I thought
I must be stupid. My teacher doesn’t really care, so…
Her main criticism is of the teaching she received:
I didn’t like the way the lessons were being taught. I don’t think that’s how I learn.
Like how the teacher used to learn me, she just used to give us past papers and sit us
down and do it and then we’d leave the lesson and she wouldn’t even mark it half the
time, so we’d kind of go away wondering is this the right answer, is it not the right
answer. … I think like because they’ve got so many people to teach I think they can’t
really spend time in the lesson writing down exactly how to do every single equation.
I think it’s more everybody has to just pick it up by themselves.
Pania (SFSP) was also demotivated by failure, feeling ‘dumb’, and not being able to get more
than a C in the Foundation paper:
[Do people think that people in the lower set are dumb then?] Yeah. Cos like when I
looked back … said I wasn’t doing the Higher, they think like oh yeah I couldn’t really
do the Higher because I wasn’t capable of doing it. So yeah it kind of put me down.
[Do you think everybody in the lower set feels like that then?] I think so because they
know that you can’t get more than a C. So you know that’s the only thing you’re
capable of doing. And mostly like … when you think about getting a C – you think
that we’re not really getting a lot.
The issue of whether they were entered for Foundation or Higher Tier mathematics was
important to many students, because of a general claim that it was easier to gain a C in
Higher Tier exams. This was supported explicitly by one of our sites, but rejected by
another, as the teacher interviews show.
4.3.1.3 Forgetting and failing to understand
Other students tended to blame themselves rather than their previous schools and teachers
for their failure to gain a C. A major theme was simply an inability to remember or to even
understand in the first place. Shelby (SFCC) was exercised by the fact that what she learned
in September would be easily forgotten by the time the exam came:
... in September what you learn, you don’t really learn at the end, you learn this stuff
every like term, don’t you? … So you’ve already forgotten it. ... you could come into
lesson and you could be like ‘Right we’re doing ...’ and then it re-jogs my memory
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from last year I think ‘Oh I remember doing that’. But if you just wrote the question
down in September and went ‘Do that’ from like July I’d be like ‘I can’t do that’.
Abbie (FELLP) said she had never liked mathematics and always struggled with it. She sees
this as a personal trait which others don’t have:
I didn’t enjoy it and it takes a while for it like to process in my head. I don’t pick up as
easy as other people ... I used to like cry before I used to go to my lessons in school,
just hated it so much.
In classes she just aims to get by without causing trouble, pretending to understand:
… when everyone else is getting it, I sort of pretend to get it, cos I don’t like to disrupt
the class or anything. … I don’t want to like drag everyone down like, having the
teacher explain to me for ages. …. I mean I know there’s probably other people in the
class who don’t get it, but we’re all just like being quiet.
Zoe (SFCP) also describes herself as someone who has never been good at mathematics and
isn’t interested in it; she can understand what to do in class but ‘it just doesn’t stick’.
I want to like it. Like I want to do good in it, but it just doesn’t, when I try it just
doesn’t stick.
Others felt that there was a ceiling on their understanding. Lucy (FELLC) says she was good
at mathematics at primary school and was also in the top set until her GCSE years, but then
couldn’t improve, and couldn’t understand:
I think I was like comfortable there... up until the last two years really. Then the last
two years I felt like I couldn’t get better… I felt like I was stuck at the same level for
the like both the two years, like I didn’t get any better. … I think I got to that point as
well like, because I wasn’t understanding I didn’t like it and then I just didn’t try.
4.3.2 Experiencing resit classes
4.3.2.1 Motivation and confidence
Most students framed their accounts of resit classes in the context of needing a grade C in
order to apply for further training in a variety of careers - nursing, cabin crew,
physiotherapy, apprenticeships – in or order to continue study - access to university
courses, health and social care, IT, psychology, and sports science. A handful said they were
simply on the course because their college/school required them to be.
For some students, this situation was a source of motivation, as for Lucy (FELLC):
I’ve had a job and everything but it’s just been a job. Whereas now I’m thinking of
like a career. … I want to do either cabin crew, do you know like on planes ... and you
need like an A to C in English and Maths to do that.
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The adult class members described themselves as highly motivated – some had come from
functional mathematics classes in the previous year, and were targeting GCSE grade C in
order to make major changes to their lives. Nancy (FELAC) was a typical case of returning to
study in order to gain a place in nurse training: after having worked for ten years in retail
before having a child, she wanted ‘to better myself’.
Some students said they were working harder now, and some even had tutors at home.
Ruqya (SFSC) described herself as different this year:
I think like last year I was the sort of person that like ... I did have an interest in
studying, but not as much. … But now I’m like really focussed and like concentrated.
… So I’m like ‘I want to do this now’.
The majority of the FE College students talked about other changes in themselves in re-sit
classes. They described being more focused and having more confidence, usually because
they felt able to ask when they didn’t understand in smaller classes with others like them.
Shelby (SFCC) is in her second re-sit class and has finally gained confidence to say she
doesn’t understand:
And last year there was more like confident people so you don’t want to say ‘Oh I
don’t understand’ do you know what I mean like? So I didn’t really like say when I
didn’t understand stuff. So then I did an exam and I got like a D in school and then I
got an E in college. And then like this year now I’m doing it again – I just tell him
when I don’t understand stuff. Like he says to me like ‘How do you feel to be one of
the smartest ones in the class?’ Because I actually ask him now and I actually
understand stuff, whereas before I didn’t ....
Lucy (FELLC) was confident that she would be able to improve in the smaller FE class:
So yeah I was like more like ... [more] chance to speak to you if you don’t understand,
cos there’s like not as many people.
Abbie (FELLP) felt more able to speak in a class of people in the same situation:
I do feel more like able to speak in this class because we’re all in it for the same
reason … But in school I couldn’t cos everybody was probably at different levels.
4.3.2.2 Relationships in resit classes
College students explicitly connected their new confidence to the different relationships
with tutors at FE college in comparison to school. For Harriet (FELLC) this was about being
treated as an adult:
… you kind of feel more mature. You know like you do more things independently
and things…. I think the kind of the tutors just like put more trust in you and they
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believe in you more … I feel like I’m more confident in college. Like in school I was
never that confident.
Joel (SFCP) tells a now familiar story of never being very good at or interested in
mathematics and not really trying until Year 10 when it was too late. Consequently he
lacked ‘the basic knowledge’ and only managed a D in GCSE. In his current class he feels
‘average’, and he appreciates the more personal touch of FE teaching compared to school.
… he’s a different person and it’s like it makes it a whole lot better because you can
actually speak to him. Like in high school and that I was never like, I was never really
one to put my hand up and say I don’t understand it … In high school it felt like if you
put your hand up everyone was just like staring at you …
Relationships with other students are better too:
No one, obviously no one was like mature in high school, whereas like here
everyone’s mature, so … I don’t feel like I’m under pressure like when I put my hand
up here. Whereas in high school like with everyone staring at you and stuff it’s like …
Like Joel, Sienna (SFCP) appreciates her FE teacher because of his good relationships with
students and his willingness to explain. She says she is now enjoying mathematics so much
that she would like to carry on to AS level.
The 6th form students had less positive stories. Imogen (SFSP) returned to the setting issue:
… in the high school it’s a lot about labelling, like not only do we have the different
sets, we have the X band and the Y band and the Y band are meant to be stupid and
the X band are meant to be a bit smarter. … I’m going to really apply myself, but if
you’re in set 4 for Maths, you’re going to think I’m not that good at this subject, so I
don’t see the point of trying. [… is it different in sixth form?] I think it can, it is and it
isn’t as well. Like I’m still 16 and basically I’ve only like left high school, so I still feel
stupid when I don’t understand something and I still feel like I can’t put my hand up
more than once because other people in the class are going to get aggravated.
Pania (SFSP) is trying to work harder and be more determined despite demotivation:
[Does it feel different being in 6th Form … does that make any difference to you doing
the class in 6th Form?] Not really, no.
4.3.2.3 Resit teaching
Some FE students talked about better teaching at College compared to a tendency to teach
from text books in school. Zoe (SFCP) prefers college because ‘in school we just worked out
of text books and like we always swapped teachers and things, so it was never like
constant’. Levi (SFCP) also enjoys the teaching at FE college:
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… the method of teaching is different from high school. Like in high school they don’t
really have fun with you or anything on the questions. They just set out the questions
and you’ve got to do it, but sometimes Mike just like goes through it and like he
doesn’t put too much pressure on us. He puts pressure on us, like challenges us to do
work, but it’s not too much pressure where we think ah, I don’t want to do this. …He
teaches us different methods to be able to do the questions. … So we can pick which
one we want.
However, despite the obvious motivation of working at mathematics in order to access their
chosen career, some students found this difficult to sustain in practice in the face of
continuing failure to understand or plain boredom. Although many FE college students
described their resit teaching as good, appreciating the smaller classes and better
atmosphere, others described it as too infrequent to really help them – Andrew (SFCC)
complained that 2 hours a week was too little - or only helpful when it was simply a
reminder. It was not able to deliver new learning of material they had never understood, as
Caterina (SFCC) explains:
… certain things that I was very good at in the past and I’m doing again, once it is
reminded or recapped I caught up very quickly. If it’s things that in the past I didn’t
like it, like ratios or big divisions I’m there like oh I don’t know if I’m going to be able,
I’m always unsure all the time.
The adults at FEL described a specific issue with teaching at FE, concerning ‘new methods’.
Nancy (FELAC) explained:
I do it slightly different, but I get to the answer sometimes quicker than others and
it’s just because I’ve been taught a certain way ... I find that easier even though
sometimes they teach you it in a different way. And I’m like I’m sticking to the way I
know, because I know that I get it right sort of thing. … it’s purely that I find it a lot
more complicated certain ways that she goes round, because on a lot of things you
can do it in totally different ways. … And I was going to say some things work for me
and some things don’t, so I’m sticking to the one that I know that works for me
instead of … I’m willing to try new things, but if I don’t grasp it straight away I’ll go
back to the old formula.
This issue came up for the FELAP students too, as discussed in 4.3.4.2.
4.3.3 Perceptions of mathematics
The students did not necessarily dislike mathematics, although many of them did; of these
latter, some said they had never liked or been good at mathematics, while others said they
had stopped liking it during secondary school. As we have seen, the dominant story was
one of being taught a mathematics which they then forgot or did not understand in the first
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place, and this was coupled with a perception of mathematics as a collection of fairly
arbitrary rules which must be learned but are easily forgotten.
4.3.3.1 Learning rules versus understanding
An emphasis on mathematics as learning rules, with right and wrong dependent on the
authority of the teacher, emerged when we asked the students how they knew if they
understood and were right. Andrew (SFCC) replied:
Because I know I’ve got it right and then the teacher says I’ve got it right and the
book says I’ve got it right and I remember the method.
For Ruqya (SFSC), it all depends on the teacher:
I will do a question and then I will just ask the teacher if I’ve got it right. And if I’ve
not got it right and he tells me I’ve done wrong, I’ll go home and I’ll practise the
questions so I make sure I know what to do. [Do you ever know yourself that you’ve
understood?] No.
Akia (SFSC) was totally focused on marks:
Because I used to do all the homeworks on time, and then my teacher used to mark
them. Oh yeah and the exam papers – at the start of Year 10 I was like really bad
with them – at the start of Year 11 there was a huge progress. Like in class everyone
used to get like … like they used to get half way, but I used to get 80s out of 100, so it
was quite a big difference. [So … so how you know you understand … have I got this
right? … it’s because you get it right?] Yeah.
Imogen (SFSP) talks explicitly about mathematics as rules to be learned, and understanding
as knowing how to apply them. She clearly thinks that there is no common sense involved:
I’ve always believed that Maths is like a game, like if you understand the rules you
can understand the game, so if I know how to do something I’ll understand all of it. …
I think it’s like if you learn how to do the equations, you know how to do the … You
can forget the equations really easily. It’s not like with other lessons with common
knowledge involved in it. …. It’s not common sense that you have to know the
specific things and then …
Some students were more explicit about understanding and wanting to understand; they
weren’t happy to just get things right, as Martha (FELLC):
I would prefer to actually know how to do it, because then I know whether I’ve got it
right or not rather than before where I just hope that it was right.
Lucy (SMC) agrees:
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Like understanding why it’s done that, why you’re like ... how it gets to the actual
answer like.
Some adult students expressed strong views about the need to understand. Maria (FELAP)
was critical of teaching ‘these days’:
I did the function skills 1 and 2 in preparation for this. [And did you like that?] Yes
and no. I found it very ‘This is what you’ve got to do to pass the exam’ kind of thing.
… Which is I think how it is these days isn’t it really? Not why you do it or how you do
it – this is the answer you need to get to get your exam marks sort of thing. … I need
to know why and how. … Need to be able to understand how. … It’s very much
calculator ... and I need to know if my calculator’s not working how I can do it.
Levi (SFCP) is unusual in his view of mathematics and how he knows he is right: unlike other
students he does not think that the teacher is always right:
… but I just, sometimes I just, if it’s wrong and the teacher is arguing that it’s right, I’ll
just say, I’ll just tell her it was wrong. Like I’ll just show her, just check over it and
show her. …. I just know it was right because if I’m working out like I just know it’s
not wrong or like I don’t know. I don’t know. It’s like saying 2 times 2 is 4. No one’s
going to challenge owt like that, but I just, when I finish I just read it again just to,
because sometimes I do make some silly, little mistakes.
The overall effect was that mathematics just didn’t make sense. Its arbitrariness meant that
it was eminently forgettable, and teaching was confusing, as described by Caterina (SFCC):
… if I do it in class I do it right and if I come the next day or two days after it’s just oh
I’m not sure, because I just really don’t know, but it’s that feeling that oh I don’t think
I did understand it or the next day he explains in different ways and you’re there like
now I’m lost because you’re teaching that way and then you’re teaching another
way.
Matthew (FELLC) tells a story about not understanding, ever. He doesn’t think that
mathematics will ever help him in life, and that a C would simply signify ‘all the work I put in
for Maths and all the hard work I’ve done trying to get the C’ . He sees mathematicians as
simply ‘Trying to create new sums for us. … Trying to make it harder’.
Some students held on to fixed ability beliefs and the idea that mathematics is hard and
only for the clever. Shelby (SFCC) said that ‘Some people are just really clever aren’t they?’,
while Ruqya (SFSC) thought ‘It’s a really hard subject though. … Cos I’ve been finding it hard
since I was small’. She thinks her older sister is good at mathematics because she has taken
after their father, and that she’s not clever like her siblings. Zoe (SFCP) also thinks that
“you’ve got to be clever to do good in it”. She sees herself as a creative person, and so
incompatible with ‘boring’ mathematics:
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I think it’s like if it was more creative and more fun and like I enjoyed it more then I’d
like it better.
4.3.3.2 The value of mathematics
As a consequence of these perceptions, perhaps, most students saw mathematics as having
only limited value in life. Shelby (SFCC) thought that no one could like mathematics since it
is so useless:
No one likes maths do they […] why would you need half the stuff you do in maths in
like life? … Like I get some stuff that’s like relevant, like percentage and stuff like
that like. If you’re trying to like work out like the percentage of your tax or like how
much is going out this month – you would need it for stuff like that, but I don’t think
you need like algebra and stuff like ... in day to day life. … why you would need like
that.
Ruqya (SFSC) agrees:
If I had a choice I wouldn’t do it. … Some of the things are useful but like … do you
know when they teach you like the circle like radius and those equations – to me that
doesn’t apply to me in my daily life. Cos I think you should just know the basic things
and not like the hard things.
As does Pania (SFSP):
somehow when it comes to algebra and stuff I don’t see how it comes to our lives. …
I don’t really see the point of learning it.
Imogen (SFSP) thinks that ‘some parts of maths connect to real life’, but:
I’ve never seen my mum using Pythagoras’ Theorem in everyday life or finding out
the circumference of something. … I want to teach ICT in universities and I don’t
think I’m going to use a lot of maths, so I think, I don’t know. It’s like it doesn’t really
click with me.
Even Yalina (SFSC), who says mathematics is her favourite subject and aspires to be a
mathematics teacher, can’t see the point:
it is very very useful for every lesson. [Can you use it in your everyday life?] Yes
I do use it … we can use it in mobile phones, we can use it in shopping, buying things
and giving money. Measuring something – it’s useful for that. [ And what about
things like geometry..] Um … I don’t think so, I don’t find it useful that thing. When
I’m doing these kind of questions I think it’s not useful, why teachers wants us to do
that?
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Where students could connect mathematics to their career aspirations, they could see
usefulness, but for the most part this was limited to particular areas of mathematics, as in
the case of Martha (FELLC):
I think it depends on the situation. I mean I need let’s say for Psychology I need, I’ll
need to be able to do statistics and things like that, so I really, so that’s going to be
quite important for me, so I do think it depends on the situation that you’re in.
Zoe (SFCP) feels that while she will need to use mathematics as a nurse (“that’s why I want
to start liking it”), that ‘angles and algebra’ and ‘stem and leaf’ aren’t useful in everyday life.
4.3.4 Experiencing RME classes
Students’ experiences of the RME classes tended to be largely positive, although we were
aware that they might be reluctant to make negative comments to us. However, we
attempted to obtain detailed comments on what students thought the aim of RME was, and
how they interacted with its methods, in an attempt to offset a general desire to please.
That this was not in fact the case is suggested by the presence of some negative responses
which are indicative of a resistance to the slowing-down effect of RME and its emphasis on
modelling and explanation. As we have seen, students’ perceptions of mathematics were
dominated by their experience of learning what to them were arbitrary rules and a sense of
its irrelevance for their lives. The RME intervention challenged this perception and made a
difference for some, but not all, students.
4.3.4.1 A different mathematics
The RME approach aims at a general shift in students’ engagement with mathematics,
enabling then to make connections and build understanding across different contexts and
problem types. While we could not expect students to articulate it in this way, particularly
after a short intervention, some students appeared to recognise that there was an
opportunity for more thinking and understanding in the lessons.
Pania (SFSP) saw RME as ‘not spoon-feeding’:
the thing I like about it cos she explains a lot more. And like she doesn’t like spoon-
feed us. … So that we can try better and stuff like that, but that’s the thing I like
about it. [Tell me more about what you mean by not spoon-feeding] Like as in she
doesn’t like give you the answer but she wants you to try it yourself and then find out
yourself … instead of her giving you all the information. […] it kind of helps us more
understand it, and then probably I remember a lot more.
Imogen (SFSP) said that the lessons ‘didn’t feel like maths’ because of the connections to
context:
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… how she’s teaching it’s not so much feeling like maths. Like a normal teacher
would probably give you a big circle and say here and make a fraction out of it.
Whereas she’s using actual everyday life things like sweets and it kind of makes you
feel like you’re not doing maths, but you are doing maths. … I understand the
fractions a bit more.
Ruby (FELLP) seemed to realise the way in which RME could capture a variety of problems,
with the added bonus of being ‘easier’:
[What’s his [RME teacher] idea as a teacher, what’s his plan, what’s he trying to do?]
Help us find ... how to find an easier way … So with the ratio tables he’s shown it that
it doesn’t matter what question you’re being given, that ratio table can help you,
because it’s used for any calculations.
Joel (SFCP) sees the RME approach as fitting in with his own way of thinking:
Yeah, she’s doing it differently from how all the teachers teach maths, but that’s kind
of like how she does it is kind of like the way that I’ve always thought about it
anyway if you know what I mean. … Just like with I don’t know just how she teaches,
that’s kind of how I’ve always like thought about it. … Like the way that she like
explains how to work it out, like that’s how I’ve thought about how to work out the
question. Like if I was doing a test then that’s how I’d think about how to work the
questions out.
Ryan (FELLP) also seemed to pick up on connections in the sense of RME as ‘mixing it up a
bit’ and making mathematics more fun:
Steve’s classes were good, yeah. I learnt like quite a lot from it, yeah. … He’s trying
to like make it a bit more like … sometimes maths is just straightforward, he’s trying
to like mix it up a bit, trying to make you understand it more. Trying to make it a bit
more fun. … And like trying to break down the questions so you understand them.
Like many others, he saw RME as supporting understanding by breaking questions down:
Breaking it down, yes. It’s obviously going to make it like a lot more efficient isn’t it?
Because if you’re just doing it all in a lump sum but if you’re doing it just bit by bit,
plus you can get more marks for that. … But like he does it like in a way where you
can understand it a lot better than …
Maria (FELAP) particularly noted the use of context: ‘That’s what I like, it’s put into a context
you can understand’.
Other students picked up on the issue of interest. John (FELLP) volunteered near the
beginning of his interview that “I’m finding these maths lessons really interesting”, and he
saw this as RME’s main contribution:
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Maths can be quite a boring subject if the teacher makes it boring, but to be honest
[the tutor has] made it really interesting, it’s been really good. … I think there’s only
one way you can teach it, it’s either … you know like algebra, you can’t do it 150
ways can you really, it’s one way. So … to be honest I don’t think he’s changed
anything, but he’s made it interesting. … He’s made it interesting so it’s made it
easier for us to learn.
Ruby (FELLP) related it all to confidence; she had found the RME ratio tables ‘extremely
helpful’, using them on every question in the post-test:
It’s a different way of teaching, but it’s also if you find something interesting and you
find you can do it you pick up on it better, you feel confident.
Abbie (FELLP) has a lot of difficulties with anxiety about mathematics but said she felt more
comfortable in her re-sit class. Commenting on the RME intervention, she says that it has
made things easier and had even made her like mathematics:
[What’s it been like?] Um ... a bit different, cos there’s different methods and stuff,
but um ... I do find some of the ways he’s taught it was easier to work out things.
…It just sort of clicks a bit more. [ So what makes it easier?] Um just like the
methods, like the bar method and everything. … Like there are questions that can be
like put down to small chunks.
She comments that it feels good to understand what she is doing because the bar approach
‘breaks it down’: “I can’t stand maths, but when I get it I actually like it. “
4.3.4.2 Time pressure and “learning a new method”
However, even though they perceived the benefits, some students said that they were
under too much time pressure to use the RME strategies in tests. When quizzed about the
bar and ratio tables, John (FELLP) said that these were new to him, and that they made
mathematics easier, but he had not used them in the post-test. He explained this in terms
of the time pressure in the test and his general tendency to ‘crumble to the floor’ in tests:
Yeah the bar was easier in the lesson when we had more time to practise it. …
Whereas in the exam we just had now, we only had like 15 minutes to do it. [so do
you mean that you’re anxious about using the time for the bar?] Yeah it’s like it
takes me like a little while…
Abbie (FELLP) was still not quite secure with using the bar; although she appreciates the bar
in class, she finds it hard to get started on her own and set up the bar model:
It’s easy when I’ve got it set up, because I know it ... then it all just all comes back.
As these responses suggest, some students tended to see RME as just introducing a new
method, a perception confirmed by Tanviha, the teacher at SFS, in the next section.
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Consequently, some were less keen on learning a new method, particularly when they felt
they had a perfectly useful one already, or simply needed reminding of a method they had
forgotten.
Although he didn’t necessarily use the RME bar himself (“the method is good, but like I
don’t really do the method properly because I don’t like doing the lines”), Levi (SFCP) thinks
that the RME approach is “helping a lot of people in the class”:
Because I was sitting next to Joel and Sienna and then when they did it first they
didn’t get it right, but then when [tutor] went over the method on the board then
they did the next question then they got it, so … Yeah, so they know how to do
questions like that now. … because I think Joel likes when things are broken down to
him like.
He sees the RME approach as being about a new method, but he is unlikely to use it, he
says:
… because her method works, but mine does too. … And I find it easier to do my
method than anything. … Like because some people when they go into the test they
just know what method they’re going to use like for each question. I wouldn’t like
want to do her method and not get it, like struggle and waste time. … I’d rather do
my method and do it fast and know that I’m right. … So I wouldn’t want to do it as a
gamble like. [How do you know they were right?] Just check over it.
Sienna (SFCP) sees positives in the RME lessons, but is not sure about the bar; like Levi, she
sees that it is helpful for other people but not for her. It appears that she prefers not to
seek explanations:
I think she is [good], like the way she brings in different things and like the things
that we’ve been doing lately I could do some things in my head, but I think the
pictures thing, but I don’t think the bar helps me with the fractions. [Okay, can you
say why?] I just think like it’s wrong to do it and like the other people in the class like
they try and explain it. It just confuses me, because I did it. I think my way is an
easier way, because I just go straight to it. … Just half it and then quarter it and then
find an extra one. … I think it’s because I can do the division and … they can’t. … It’s
just when, I only find it confusing when the rest explain it and like they try and get to
the answer and then they’ll be like finding half and they have to add another one
when they could just do a division and then it would give their answer.
Like other students, she sees the aim of RME as providing an alternative method:
I think she’s trying to give people different ways or like look at it with a different
perspective and if like they struggle one way they’ll have other ways to do it, so when
you get in the exam if you forget one you’ve got another.
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The adults at FEL presented a particular case of this issue, which appeared to be based on
their frequent experience of finding that mathematics teaching was different from when
they were at school. For example, Maria (FELAP) echoed Nancy’s (FELAC) preference for
‘going back to the old formula’:
When they first came in ... I think it’s cos I had a system in place already that I used
and we were being taught a new system, I struggled, and I kept sort of brushing the
new system aside cause I had a way of doing ... yeah, cos I already had a way of
doing it, I couldn’t get my head round their way of doing it. The way I was doing it
was very similar, but without drawing a bar and stuff like that.
She acknowledged that the RME tutor had said this was OK:
But he did keep saying to us if you’ve already got a system in place use it. This is just
a system if you can’t do it.
However, she was more than happy to change when she no longer felt confident about the
mathematics:
Because I don’t understand how to solve these problems, I’m enjoying it and
benefitting from it a bit more, if that makes sense.
4.3.5 Summary
Many of the experiences of mathematics teaching related by the students were familiar,
echoing the literature on the impact of traditional teaching, setting, and perceptions of
mathematics as a set of rules to be learned. Students who had entered FE College from
school reported on more positive experiences in terms of relationships and confidence,
although they recognised the impact of having too little teaching time. For the intervention
students, RME presented something new, which was welcomed by some as giving them an
opportunity to understand and to break problems down. However, some students saw RME
as just about learning new methods, which some resisted because they did not want to
replace existing working methods. This was a particular issue for the adult students,
although they could also see the benefits of RME, particularly when they lacked a current
strategy or did not understand.
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4.4 The teachers’ view
We interviewed teachers in all four sites about the challenges of teaching GCSE resit classes,
their approach to teaching this particular group of students and how they would like to
develop their teaching in order to meet their needs. Four teachers were host intervention
teachers, and we also asked them for their opinions of the RME approach.
4.4.1 The challenges of teaching GCSE resit
4.4.1.1 Confidence and motivation
All the teachers identified issues of confidence and motivation as the primary challenges in
resit classes. Some noted that previous teaching had been patchy. For those in the FE
colleges, attendance was an additional issue. Mike (SFCP) sums it up - students arrive ‘with a
multitude of problems’:
It’s not uncommon for them to be saying they’ve had half a dozen teachers, they’ve
had supply teachers. All those issues affect their enjoyment of the subject, their
confidence in the subject, so our issue is … two classes of people who don’t want to
be there, have a lack of confidence in the subject, often don’t like it. Having said that
the vast majority do have the maturity to recognise they need the qualification, so
we don’t particularly have issues with behaviour. We will have issues with apathy to
a degree, lack of confidence and willingness to work particularly hard, attendance,
punctuality sneaks in as the year goes on. Those are sort of the issues we’re up
against.
David (SFCC) noted that motivation was the key issue in the classroom, keeping students
going who feel they ought not to be in the class because they have narrowly missed a C, but
also those who feel demotivated because of persistent failure:
… keeping them concentrating, keeping them focussed because at the other end of
the spectrum it’s just a trial for them every week and they’re trying to do things that
for the last four years at school they didn’t understand. Why should they understand
it again in the next year?
He sees confidence as the major challenge for resit students:
Because when we’re teaching the subjects, if we do something like Pythagoras, if we
do something like something like bar charts, but pie charts for example, “we did this
at junior school … we you know we’ve done this”. … but they don’t know it. … And
it’s that, that’s the barrier. That’s the confidence barrier because they know it and
they sort of they won’t listen, because I think they’re afraid a little bit of listening into
bits that they don’t understand, so they put up a front of “I know this already” …
Tanviha (SFSP) sees the problem as one of needing to move on:
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Do you know what it’s motivation. I think the key thing is trying to get them to not
think about the grade that they had back in the summer, but start afresh and then
try and work towards the C that they need … I think once we’re over that, then it’s
okay. I mean it’s trying to get them remember, reconsolidate, but the past things
that they remember and how they apply it again, the hardest part is “oh we’ve just
done this again and we’re doing this again and I’m never going to get it”.
Kate (FELLC) thinks that lack of confidence leads to non-attendance:
The main challenge has been attendance. … they are reluctant to address I guess the
difficulties they’ve had with it and it lowers their confidence, at which point they
decide not to return because they’re not enjoying it. When they’re in the classroom
the difficulties with them tend to be that they are down on themselves about it, so
they tend to feel that no matter what you know they’re never going to get it.
Like Kate, Carol (FELLC) sees attendance and confidence as part of an overall challenge,
particularly within the college framework of three hours once a week:
Well bearing in mind you only see them once a week, so there’s no, you’ve got to
keep them very positive, very encouraged and motivated and also tight, so they
come, they have friends that come every week, so for me my classes have to have a
certain element of enjoyment in it as well, social and you know interaction and
enjoyment. Otherwise these people just won’t come.
Peter (FELLP) makes the same point:
Yeah, and they may have got less than a D the last time, so they’ll have a few ‘go’s
and they will achieve a D, so confidence in using Maths, attendance, normally just
their tiredness towards the end of the three hour session as well. It’s a challenge.
4.4.1.2 Basic skills
Aside from these issues, the main challenge was addressing the very low base that some
students are starting from. Kate (FELLC) noted that “you’re talking even looking back to
entry level 1, 2 and 3”. Lucy (FELLC) is also concerned about students’ study techniques, and
their lack of experience in taking notes for themselves, echoed by Asad (SFSC), who sees
students’ lack of independent learning skills as a major obstacle:
.. they’ve got to be, learn to be independent learners and these, most of the resit girls
are developing that independent learning. They’ve not mastered it. If they would’ve
mastered it, they would’ve been in set 1 or set 2 and things like that, whatever,
because they’re capable of working individually for several hours at a time, because
they feel comfortable about it. Where we’ve got girls who are not individual
learners, who don’t have the acquired skills to actually revise, then they still get
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flustered, they still don’t know how to revise, then we’ve got a repeated pattern of
failure in revision ...
Mike (SFCP) pinpoints a lack of basic skills and failure to retain as a major problem, which is
exacerbated by the time pressure of the course:
The positive is they have seen everything before. … They do have an idea. Your
problem comes when you get students who can’t do multiplication, who can’t do
division, because we don’t even put that on the scheme of work. We haven’t got
time ... to go back. We haven’t got time to teach them their tables and the number
of students who are, I’ve got a times table chart on the wall, who are looking at the
wall or you see them with their fingers. … Because we’ve condensed everything that
they do in Year 10 and 11 into about 30 weeks as it is. I think virtually everything we
do or everything we do, the majority of them will sit there and can do it on the day. …
The trouble is if I reintroduce it a fortnight later, the vast majority have forgotten.
Peter (FELLP) sees slightly older students as more motivated and willing to work and take
notes, but he notes that there may be pressure in the future when all students arriving with
a D must do the course. He believes that motivation will dip, and he feels that there may be
a particular problem with girls.
But the 17, 16, 17 year olds …Their group thing together, they bring each other down.
They talk each other down ... in the classroom and you can see it happening. They
out compete each other to be stupid in Maths
Tanviha (SFSP) is the only teacher to point to language issues, which are very evident in her
class because some students are not first language English:
I mean if the girls can’t access the question and don’t understand what’s going on, if
it’s too wordy they’ll just give up. … They won’t even try to attempt to even read the
question and then have a go, so it’s kind of like getting them to understand that why
don’t you highlight the key points? And then take it from there, but with the resit
group I think it is just the motivation.
4.4.1.3 Mixed abilities
A related issue was mixed ability. David (SFCC) sees this as the main challenge:
I think the main challenge seems to be confidence, because we have people who just
missed a C and we’ve got people who just got a D and it’s a big range and so that’s a
range of arrivals, but also there’s a whole range of there’s the people who were lucky
just to have missed a, you know got a top end D and people who were unlucky not to
have got a C, so even at the top there’s sort of people who did it by cramming and
who did it by ability and then I think our challenge here is trying to, although they’re
all D grade, which is a great advantage, I think the disadvantage is sorting, well the
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difficulty we have is sorting out who needs to be just improve their practice, so
they’re confident they’ll pass, and others just telling them yeah you don’t have to
check it at every step of the way.
Asad (SFSC), also picks out the variability in the class, and relates this to the school decision
to teach Higher Tier, when nearly half of his students have been doing Foundation.
Now this is a departmental thing and what the department has decided is that
because the results, 85% of our GCSE passes came from Higher, okay and they’ve
decided what’s the point of doing Foundation? And they’re going to fail, because the
majority of the pass mark for grade C as you’re aware it could be as high as 75% on
the Foundation. …. So we’re talking about the very low Foundation and suddenly
they’ve got to do Higher.
He has struggled to get to know his class of 23 girls and to address their varied levels of
attainment:
If it was like a smaller group it would’ve been manageable, but that is one dilemma.
The other dilemma is we’ve got girls who just missed out on a grade C, so we’ve got
this whole differentiation from girls who don’t know how to half 64 to some girls who
want to do in terms of I don’t know some topic in Maths and things about whatever
you know, and they can do plotting a quadratic graph quite comfortably.
4.4.1.4 Time pressures
Consequently, time was a major issue for them as teachers. Kate thinks that although
theoretically the resit course can fit into a year, time is an issue because of the amount that
needs to be covered and the need to consolidate learning and support individual students:
In a year having three and a half hours with students it wasn’t a problem, but that
was broken over three days in [her former 6th form college]. Here it’s one night a
week and I’m not sure if that’s going to have a big impact …. they’re only seeing me
once a week and they haven’t got the support of me sort of any other time. Like I’m
not seeing them, sit down with them and say this is how you could do this more at
the time. I think that might be a consequential difficulty…. Two and three hour blocks
… loads of concentration on so many topics, it can be a little bit too much. … You
know it’s quite a sizeable group if you’re looking at people who need individual
attention, so … they’re not getting what they need ... to really make a big
improvement.
She estimates that just 40% of her class can be certain of achieving a C, largely because she
has insufficient time to get to know her students and target their strengths and weaknesses.
Lucy (FELLC) also thinks that the college’s class scheduling of one three-hour class a week is
problematic:
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I don’t think you get three hours’ worth of work in three hours … I mean the other
option would be to give people homework, … I’ve been giving a little bit, but I think
people won’t do it, so you need to make sure you cover a lot in the class I think,
because they won’t do it you know. They’ll walk away and they’ll just get the
materials out the next week sort of thing. I know that, but I do think you know it’s
very difficult to get that amount of stuff in the three hours.
Asad (SFSC) thinks the major problem is the pressure to achieve a C in a short time:
Teaching it’s all about equipping them to, in terms of life and everything, it is a
problem solving activity and things like that, whatever, so they will have in terms of
the aspirations, big aspirations from those who want to do up to degree level, things
like that and whatever and maths comes up in things like that, whatever. Then
you’ve got people who just want to go to work, work environment and everything …
but the question is are we preparing these girls to a work life and everything? The
answer is maybe and everything. Unfortunately in all school systems it’s the same.
There is a tick box exercise. You’ve got to get the grade Cs regardless. Now
unfortunately we can’t teach them thoroughly in terms of basic sale price,
percentages and things like that whatever, numbers and everything. We’re
cramming all this information into their heads…
He sees current work plans as impossible:
If you look at the .. first week I’m doing decimal places, standard, sorry significant
figures, recurring decimals and upper and lower … all in one week to this group.
Okay, next week I’m going to do fractions and so that is more or less telling me I’ve
got to do one topic per week, sub topic. And everything, so in the space of two weeks
I could’ve covered six topics. It’s ridiculous.
Consequently, the one-year course is untenable, in his view:
The girls, if you’re going through so many topics, it’s a two year course higher
they’ve never done before, cramming into six months, I will cover 40 topics in six
months and everything. Will they remember it? Good for them. That’s not going to
work and things like that, whatever. Going back to the question, is the system right
or wrong? It’s not, of course it’s wrong …
4.4.2 Approaches to teaching
Teachers tended to talk primarily about changing students’ attitudes to mathematics and
motivating them to learn.
4.4.2.1 Addressing confidence
David (SFCC) addresses the confidence issue head-on:
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Well I just have to, well I apologise to them and then I say just bear with me because
not everyone gets this and I turn the emphasis on the fact that people have got a D
here. You’ve all done really well. You’ve all got a D. That’s great, but the fact is the
reason you didn’t get a C is different from the reason that he didn’t get a C and
there’s only little bits. We’re not looking at getting, you know if you got, if you fell
five marks short of a C, because most of them seem to know what, we’re not looking
to get six marks. I’m looking to get another 20 marks, so that you’ve confidently got
a C and the things that you need to get those extra marks, there are only a few marks
on each subject, and it’s because there’s just a few little things that you don’t
understand on each subject. The problem is you’ve got to listen to everything
because otherwise you’re going to miss the bit that you didn’t understand last time.
Kate’s (FELLC) main priority is confidence building, which she addresses by breaking tasks
down and situating them in story contexts and real life examples. She also encourages
students to work together:
I think it gives them the ability to explain the methods and give each other different
methods which they’ve not got before and it helps them communicate in language
that they prefer rather than just hearing it from me.
Peter (FELLP) also tried to ‘mix things up’ with group work:
I try to make things active for them, so we’ll switch what they’re doing quite
regularly, so I try not to keep to a routine so to speak. … So they may work in small
groups and they might discuss things like, later on we’re going to start handling data,
so they’ll be discussing what words mean … They’ll be doing exercises and practising
things as well. So all those things. I try to mix things up as much as I can.
Like Peter, Lucy (FELLC) aimed to do group work, partly as a diagnostic strategy but also
partly in order to offset the problem of the three-hour class and enable students to engage
more with the material. Although she aims to avoid too much lecturing, she does feel the
pressure of time:
I think that always unfortunately has a, it can have a bad effect in the way that you
think “we’ve not got time to do this, we’ve not got time to do” … you know because I
actually I really like the idea that people can learn by investigation, but a lot of the
time it gets stunted that, by the fact, by the you know the time schedule that you’ve
got.
4.4.2.2 Changing attitudes
They also attempted to change students’ perceptions of mathematics and mathematics
learning, focusing on relevance and the value of mistakes. David (SFCC) tries to make
mathematics relevant:
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if they know what they want to do, you can pull out things and say you know ratio
you’re going to need this because you’re going to be mixing medicines or if you’ve
got a ward of, not a ward of, if you’ve got a set of patients, of 10 patients, it’s going
to be a bit different from if you’ve got a set of five or two or whatever. … you can give
them a bit of application and if they’ve looked into things, and also by now when it
comes to percentages for example you can always find someone who’s been, got a
job and is working so you can pull out percentages and the value of their sort of
overtime you know and the value of a rise, that sort of thing.
Like David, Lucy (FELLC) also tries to tailor her teaching to the other subjects students are
taking in order to put the mathematics into context.
Yeah, I mean I’ve asked quite a few of them already what subjects they’re doing, so I
know like someone’s doing motor vehicles, someone’s doing media. The new girl
today she said she’s doing hairdressing, so when I do the examples and things I bear
in mind that …
Carol (FELLP) and Peter (FELLP) both try to highlight the value of failure; Peter explains:
I will try to emphasise the value of mistakes and the value of self-esteem in the
classroom, so I’ll promote mistakes and we’ll all learn from someone’s mistakes, but
that’s difficult because you know it’s like, it’s not, I don’t want to make it, you know
there’s a balance between showing somebody that’s a really interesting mistake to
make, because everyone will make that mistake and then all of a sudden their ego’s
dropped and “I’ve made the mistake and oh no, no. I’m never going to get this. I
can’t do this” and everyone’s joining in.
Unlike some other teachers, Tanviha (SFSP) says that her students are asking for something
different:
I don’t want to be where they’ve been like five years and teaching something on the
board and just teaching them a certain way. I think they learn better if they’re telling
each other how they’ve done a certain part of the question and then bringing that
question together and we’ve done that today and they worked really well, so I got
girls who are of similar ability to sit together and then work on a question and then
they’ve just put the input in what rules they’ve used, because they’ll all bring in
different rules. … But it is hard…. It’s draining yeah. It’s draining yeah.
Like Asad (SFSC), Tanviha (SFSP) has a wide range of ability in her group, and has to work
hard at differentiation:
With the work in this group, because they range, because I’ve got some Gs in there,
I’ve got Fs in there, I’ve got Es in there, so it is quite, even though it’s the foundation
I’ve still got quite a bit of a range in there, so they’re all split up, completely
differentiate things. I’ve got a lower group doing, on the same topic, but different
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type styles of questions and another group doing a different type of question and
another one trying, tackling more harder, think about the D and C grade questions.
4.4.2.3 Targeting skills issues
The teachers also aim to address skills and target gaps in knowledge, but there are limits to
what they can do, as their comments on the overall time frame show. Kate (FELLC) has
learned that she needs to ‘just get cracking with number work straight off’, while Mike
(SFCP) talks about targeted ‘fixes’. However, even with a slightly more homogeneous group
in comparison to Asad (SFSC) and Tanviha (SFSP), he struggles to do what is needed, and
sees this as being more difficult once the college is taking students with grades below D:
… we’ve got a problem in as much as you’ve got a one year revision course. We don’t
have the time to take apart what they couldn’t do and put it back together. … It’s
worked relatively well, okay, up to now because again if the people have got a D they
do have a decent grounding. What we’re trying to do is we’re trying to find three or
four things over the year that we can fix. Um, again that may not work so well in
years to come where you’ve got weaker people who you can’t do the quick fix with
and we could do with more teaching time. … Which is why it would be better if they
were separated in the future, we could do this over two years. Um, so you know
much as we would like to go back and dissect what they’ve learnt or not learnt at
school we don’t have time, so it’s looking for quick fixes.
Consequently, the system relies on time for revision and past papers immediately before
the exam, since retention is generally poor:
And we have them in here from 9 til 3 and they’re on a carousel where each teacher
teaches the same lesson maybe five times and the students start in one room and
then after an hour they get up and go to the next room and we pick five major topics
and try and blitz them as a revision the day before. … From what I’ve said before
from a retention point of view they will have forgotten it and I also know that an
awful lot of them won’t do any revision anyway, so at least they’ve had four or five
hours the day before.
Retention is a major challenge, so Asad (SFSC) optimises consolidation:
Now some of those girls, even though they’ve done it they still can’t do it. … it’s just
like it doesn’t matter how many times you say it, it doesn’t click in and everything, so
… one thing that works very well is if I do one topic and repeat it three times
throughout the year, it does help. … Even though I try and do it as much as possible
to the extreme and everything, but when I find though it’s getting too much, then I’ll
stop it. Then I’ll repeat it again, perhaps add a little bit more and everything. That
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way they’ve got this certain oh yeah, they’ll always remember a certain percentage
of it.
Despite his efforts, there is not enough time to address all the issues:
I have changed my teaching because I am more concerned about getting them
through and getting that grade C, so I am targeting topics which occur regularly and I
even wrote down a list of ten topics for this November resit. You know target this,
this is going to help you. You, some of them perhaps you know that, but this is going
to help you to get through and everything, so it’s targeted work and unfortunately if I
were to do it thoroughly and properly there’s not enough weeks.
4.4.2.4 Higher versus Foundation Tiers
As we have seen, there is an issue over whether students are entered for Higher versus
Foundation Tiers. Asad (SFSC) describes teaching approaches in terms of the school’s
strategy for meeting exam targets, including a switch to Higher Tier for all students,
together with a decision to enter some for November resits. Teaching had involved going
through past papers whether or not students were entered for the November sitting, but he
had made the decision to split his class into students who were likely to pass and those who
were not. He had begun to work on individual topics with the latter group and had
introduced extra lessons after school:
I think nearly the whole group, have done Foundation and suddenly they’ve got to …
…So really it’s just selecting the topics and saying right obviously we’re going to stay
away from the A* and the grade A topics, because it’s way beyond them. However,
some of the Foundation topics come up in the Higher, like the numbers, you know
highest common factor, lowest common multiple, things like that, whatever. They
come up all the time. Straight line graphs comes all the time in the Foundation and
the Higher, so it’s important to cover those topics again and everything…
The issue of Higher versus Foundation Tiers has also come up at Mike’s college, with many
students believing that they have better chances in the higher paper:
Um, our argument has always been the level to get a C is the level to get a C. There is
that crossover on both papers. It is the same or as near as makes a difference the
same standard. If you need more than a C fine. We will talk to you and we will do
some extra work and let you do the higher paper or see if we can do that. If you’re
going to go for the C we would rather you did a paper where you can do everything
rather than go in and doing a paper where you can only do half of it and you now
only need 30%, but if half the paper’s gobbledygook to you it’s 30% or 50%. It’s not
so good you know.
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4.4.3 Teachers’ views on the RME approach
4.4.3.1 The positives
Teachers were very positive about the RME approach as a pedagogic strategy. Mike (SFCP)
saw it as radically different from what students were used to, with far more potential for
understanding:
Right I think the power of it is if you are trying to … you’re trying to teach a method
where you’ll say to them … say if you think of things like the bars, you’re trying to just
say to them ‘Right what information do we know? – put it down in this bar and see
what else we can work out from that.’ So it is a more open general approach to
problem solving, which I would hope people could apt to different environments. If
you get used to that ‘What do I know? What else do I know? Where am I trying to
go with this? Can I put those combinations together?’ – as opposed to the standard
way that it’s taught in this country. We teach it here as well … ‘If you get this
question you do this step, that step, that step’ – and students are drilled to do that –
which works for 70, 80% of questions. As soon as you get a slightly different one …
or you forget. … I think [RME] potentially would give you a foundation you could fall
back on and apply to lots of areas. But that is really hard for us to get over to our
kids on a short block. What I hope it does do with them is it opens their eyes to
different ways of dealing with things … and in some cases it switches a light on.
Carol (FELLP) focuses on RME emphasis on drawing diagrams as enabling students to ‘give it
a go’ and gain confidence:
… seeing things in pictures and making sense of it … if you’re stuck just draw
something and it’s that idea of visualising it rather than thinking oh my gosh I can’t
remember the rules here … So that’s been really effective. It’s also really good for
giving yourself, I think what my students like to do is it gives them space to muddle
through it rather than oh my gosh, what’s the rule? And not knowing where to start
and what their next step is, so it’s given them a bit of a strategy and a bit of a process
even if you don’t know where you’re going initially, you can by drawing get there. …
It takes a lot of the panic away. It takes the anxiety away and it helps you, even if
you don’t know the answer you know that you’re going to be able to find it if you
keep drawing things.
Like Mike, Peter (FELLP) saw a major virtue of something different from school:
Um, I think it’s very good techniques for them to grasp onto, because it’s different
from what they’ve been doing at school … Yeah, so having another one like ratio
tables or whatever and using the grids is a good tool to use, so if they have a raft of
tools to pick from and they’re not just staring at the paper thinking I can’t remember
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how to work out the percentage of one or another, it’s somewhere to start at least
isn’t it?
Tanviha (SFSP) has used the bar in her Year 7 class, and is enthusiastic:
I used the same method with my Year 7 and I started this last week with them where
I introduced percentages and I liked it and it did, they understood it and they could
see it better. They realised how to work out 1% without me even telling them how to
work out 1%. It was really good. It was really, really good and I can see it, I mean
they were a bit of a middle, no actually they were one of the top bands, but I can see
how it could work with a lower ability group, because the lower ability are much
harder to get them to understand what’s going on…. I think it’s really, really good in
the sense of is it more looking at what’s happening instead of just numbers
4.4.3.2 The challenges
However, Mike (SFCP) notes that some students, particularly stronger ones, resist the RME
approach because they are reluctant to learn a new way of doing the easier problems which
they can solve and consequently don’t see the point. However, these same students can
benefit in the long run:
Because some of the better ones that we have in these classes when [RME teacher]
has done it, and it’s all relative obviously, but some of the better ones do sit there
and say “well I can do that anyway”. … But in terms of an approach that breaks
things down and gives you a hopefully a standard way that you can fit lots and lots of
mathematical problems into, it’s really, really good, because a big issue we have is as
soon as a question goes slightly off the basic that they’ve seen … they’re stumped. …
So if we have a framework that you can adapt to a whole raft of problems, that
would be really, really useful.
Mike was quite concerned about student resistance, suggesting that the RME approach
needs to be introduced early in schooling:
I think the problem with dealing with not just the students here, but certainly
students of any age in this country is that they have pre-set ways of doing things. By
definition the students I’m teaching are not outstanding at maths and there are
weaknesses within their thought process. But neither are they massively poor either
if they’ve got a D. That does mean in some areas they’ve got structures that work. In
other areas they’ve got structures that don’t quite work but are deeply embedded –
it’s a problem. To change that mindset and that thought process with some of them
at the age of 16, 17, 18 … is quite awkward. So presenting them with a new way in
some cases is met with a ‘But I know how to do this’ is the issue - even if they actually
don’t know how to do it. So you’ve got some structure … which as I say in most cases
does have probably some substance to it, but is maybe weak around the edges … So
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sometimes it is legitimate ‘I can do these questions, I don’t need another way’. The
process itself I think, if it as embedded in younger students in this country I think is
extremely powerful. I think the visualisation of what’s going on and the ability to
adapt it to different areas … cos our students struggle like mad to adapt anything -
give them a process and a standard question that fits that process and you can drill
them to do it. Go just slightly off that path and they’re unable to adapt it and to
apply it.
Peter (FELLP) raised the same issue:
… so some of them will have this thing where they’ll say just tell me what the answer
is or tell me what to do, tell me how to do this. … And they don’t like that … but I will
tend, like Steve tends, not to give answers. He will tend to ask questions and that’s,
for some of the learners that will frustrate them because they just say “tell me what
to do”.
4.4.3.3 Pace
As a host intervention teacher, Mike had to give class time up to a slower pace and
coverage. Early in the year, in the number module phase, he felt that this was not a
problem, because there was some slack in the timetable:
And touch wood I’m on target. I don’t need that last week at the moment. Um, you
know you can rattle through things. I can afford to lose a small number of lessons ...
However, later in the year he was feeling pressurised:
With every other group I am three or four weeks ahead of [the RME one] and where
am I going to squeeze in this and this and this? But you’re right about the underlying
understanding being really really important, so I’m pulled two ways. … I really like
what you do and buy into it, and the other side of me is saying ‘damn, with this group
I’ve still got to cover this this and this, and when am I going to do it?’, because when I
start teaching again I’ve still got things on the scheme of work to do…
Mike’s view is that the time pressure is a major problem, when weaker students probably
need a two-year course combining an RME approach with recovery of the basics – “if we
had two years where a good chunk of the first year was going back to basics and just
dismantling what she knows or doesn’t know, or thinks she knows, and putting in some
structure … some way of dismantling all the misconceptions and rebuilding the basics”.
In terms of the impact of the intervention on her as host teacher, Carol (FELLP) echoes Mike:
Well it is a challenge definitely, because what we have done is spent a lot of time on
the number part and I think that’s really useful, but it just means the rest of the
course is quite squeezed, so the pace is good. I mean it would be nicer if we had
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double the amount of time really, because we have to get them, or Paul and Sue have
to get them to a point of making this sort of workable quite quickly. Whereas in a
sort of three classes a week at school they have a lot more time to develop it you
know.
Despite these pressures, she thinks that the intervention will benefit the students:
… the good thing is when we do this, the making sense stuff, you cover it very
thoroughly, so your students are definitely going to get marks in those questions, so
you know and it pervades into so many other topics that I mean it’s genuinely worth
the investment.
Peter (FELLP) is comfortable with the slower pace of RME because of the longer-term
benefits:
Yeah, I think once people have got it they can move through it quite quickly. You can
apply things across the board can’t you? …So like ratio tables, once people have
learnt that, if they can see it works elsewhere then they’ll use it elsewhere.
Tanviha (SFSP) sees similar problems for resit groups as other teachers in terms of
introducing RME in a time-pressured frame, when students are anxious to work more
obviously on exam questions:
… at first they understood what was going on and then after a while they felt like
why are we drawing bars for? Why do we keep drawing bars? They didn’t
understand what the bar represented. … They just felt like it’s another method.
…They didn’t realise that it was a skill that they were picking up. It was just a bar,
because some of them just drew a bar in one of the questions and then they didn’t
know how to apply the question to the bar. … So that’s where they were. They just
drew it and they’re like okay, what do I do next? So they didn’t realise it’s not a
method. It’s the understanding of what the whole bar would represent and what
would you do next?
Tanviha feels that the RME approach is valuable, but comments that the resit students need
to understand more about what it is doing in order to accept it:
So you want to try and cover or get what they’re really weak at. I mean number is an
important part and they said “oh it’s like we’ve been doing bar for the past four
weeks”. “In fact actually no you haven’t been doing a bar. You’ve been doing
fractions, amount of fractions, you’ve been adding fractions, you’ve done ratios and
you’ve done percentages. That’s five different topics you’ve done in that space of
time and it would take me a lot longer to do, so they didn’t understand that
concept”. They just thought “we’re just looking at a bar all the time”, but they didn’t
realise. I think next time what it has to be is them to realise what topic is being
related to. Maybe then they would understand “oh I can use this in fractions, I can
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use this in percentages, I can use this in ratios, I can use it”… because they just, as
they would walk in they’d see a table or a bar they didn’t realise what they were
using it against.
4.4.4 Summary
The legacy of students’ previous experience of learning mathematics presents particular
problems in terms of (1) their tendency to understand the RME approach as ‘just another
(algorithmic) method’, and (2) their resistance to, or lack of belief in, sense-making in
mathematics. However, while the RME approach increases the potential for success, it
changes the nature and pace of the work, leading to the possibility of resistance from
students (especially those on the C/D border). Teachers’ views were positive, but they
expressed concerns about the time needed for RME in the form of short interventions in an
otherwise very different programme of work, and also the extent to which students are
likely to engage with it. All see it as an approach with great potential for weak students, but
as presenting problems for stronger students who might reject it. Teachers were strongly in
favour of introducing RME earlier in school careers.
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5 What have we learned?
5.1 The post-16 landscape revisited
Realistic Mathematics Education prioritises use of context and model-building to engage
and motivate students, enabling them to visualise mathematical processes and make sense
of what they are doing without resorting to mis-remembered rules and procedures which
have no meaning, a problem noted in much 14-19 mathematics provision. This problem is
exacerbated in short-course GCSE resits, and targeted, meaningful content is needed to
break cycles of failure. RME has the potential to raise the self-efficacy of students who have
experienced long-standing failure in mathematics, and consequently enhance their
engagement, understanding and attainment. However, this is a difficult – but not impossible
- context in which to introduce a new approach, presenting challenges for students,
teachers and researchers.
5.1.1 Students
Resit students have a history of repeated failure in mathematics, and many have limited
basic number skills; they are particularly wedded to remembering a variety of formal
procedures, which they nevertheless struggle to retain and replicate. Their confidence levels
tend to be low, and they have little sense of authority over a mathematics which they
generally do not perceive as having anything to do with common sense. Many are desperate
to achieve a pass in GCSE mathematics which will enable them to meet their career goals,
and this is a source of motivation for most, although some find this difficult to sustain in the
face of not understanding. Some of the 17 year olds believe that if they can ‘brush up’ on a
few topics then they will achieve success, and this can make them very reluctant to engage
with a new approach, particularly when it presents a mathematics which is very different
from the formal algorithms they wish to revisit. Used to being taught a quick route to formal
procedures, they are challenged by being asked for their opinions and strategies, and to
engage with other peoples’ solutions. They need time to adjust to this approach and to shift
their expectations of what a mathematics lesson looks like. However, as the findings
indicate, they did employ RME strategies to good effect, and they were able to engage in a
different kind of classroom interaction. Some students clearly appreciated the RME
approach as a way of helping them to understand more and make connections. The adult
students also tried to lean on old remembered formulae, but these were less likely to be
available due to the passage of time, and they were open to new methods. They also
expressed a desire to understand, and the RME approach was met with enthusiasm by many
of these students, some of whom recognised the power of the models to unify topic areas
and significantly reduce the reliance on memory.
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5.1.2 Teachers
Teachers of GCSE resit feel a great deal of pressure to cover as much of the syllabus as
possible in a very short period of time, and this can lead to an approach which involves
quickly revisiting the formal aspects of a topic, even though they know that this is going to
have a short-term impact and is unlikely to be successful in the long run. They are under
pressure to produce results, and are aware that this forces particular choices. They are
acutely aware of the difficulties their students have, citing confidence, motivation,
attendance, and skills gaps as major problems in learning mathematics and recognise that
repeating a traditional transmission and drilling approach is unlikely to be successful. While
the host intervention teachers were anxious about pace and student resistance to RME,
they were enthusiastic about it as a way of teaching mathematics which could give students
a chance to learn. They were keen that students should see the benefits.
5.1.3 Challenges for the intervention
As researchers and tutors in this project, we had an in-depth knowledge of the theoretical
framework behind RME and also a working practical knowledge of how to implement RME
approaches within the classroom. We were aware of the need to spend time embracing
contexts, developing students’ informal representations and removing ourselves as the
mathematics expert in the room. We had a vision of the classroom cultures necessary to an
RME approach, the need for students to think independently, to explain their strategies to
the class as a whole and to question each other’s’ approaches. Post-16 GCSE resit students
are more reluctant than most to expose their thinking and to engage with the mathematical
ideas of their peers. Finding ways to engage them in a shift of classroom norms proved
challenging and caused a particular tension for us in relation to time. Had the intervention
been longer we would have spent more time developing the classroom culture, but the
confines of a limited number of hours teaching, combined with the concerns expressed by
teachers about covering their schemes of work, necessarily led to compromises on our part.
5.2 Suitability of the RME approach for GCSE resit
5.2.1 Effects on student outcomes
The independent evaluation (see Appendix 5) concludes that the RME approach had some
benefits despite the limited nature of the intervention. It recognises the challenges for
implementation in this context but suggests that this is an approach which is worth pursuing
in further trials. Our qualitative analyses show that there is potential for impact, and suggest
that this would be enhanced by different time frames for GCSE resit classes, in particular
extension to a two-year course.
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5.2.2 The materials
The emphasis on use of context in the materials interested and motivated the students. At
times group discussion became lively, and spontaneous, as various students offered their
opinions and life experiences. Contexts such as the ribbon and the computer download bar
encouraged students to make sense of the strategies they used.
The RME models associated with the number module - i.e., the bar model and the ratio
table - were shown to be particularly useful for GCSE resit students. They provided them
with a new way of representing and thinking about a variety of familiar problems; they
promoted informal strategies which can be very helpful to students, particularly those
around grade E and F level; and they provided a visual representation through which
students already competent in formal procedures can gain insights as to why their
procedure works. The RME models are ‘new’ methods but they are not necessarily trying to
replace the ‘old’ ones. They unify many elements of the number curriculum, which is
particularly important within the confines of a short course, and they have the potential for
use within other strands of the curriculum.
The algebra module offered less obvious cohesion than number in that the models
associated with algebra were more varied. One of the algebra topics focused on how to use
the bar model for solving a variety of word problems. Unfortunately, due to time
constraints, this session was only trialled with one of the four groups, but we have since
developed this material as part of a CPD package and it has been well received by teachers,
who have found it empowering to realise that the bar model has potential to develop
approaches in algebra as well as in number. The pressure to move from contexts which
develop pre-formal algebraic thinking to formal symbolic algebra was more apparent in the
algebra module, not least because this was a shorter module.
The materials are made up of a variety of genuine problems and as such they make a case
for enabling students to develop their problem solving abilities as an integral, rather than
‘bolt on’ experience. This is particularly valuable in preparing students to sit the new 9-1
specification GCSE with an increased emphasis on problem solving and contextually based
questions.
5.2.2.1 A note on algebra
The intervention group performed at a lower level than the control group in both pre- and
post-tests in algebra, although at a higher level on the delayed test relative to the control
group. Here we consider the issues and the successes relating to the algebra materials in
more detail:
The Algebra module was relatively short (9 hours) and in reality even shorter: two
classes received 7.5 hours, another class 4.5 hours. This meant that tutors had to
attempt to make short cuts through the materials or miss out sessions altogether. Such a
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short intervention is not conducive to an RME approach in terms of allowing students to
genuinely embrace contexts and to gradually develop their informal representations.
GCSE requires that students reach some demanding formal levels of symbolisation and
algebraic manipulation, and in some cases the gradient of moving from the RME
contexts written into the materials through to answering GCSE-type questions was too
steep. This issue was exacerbated by the reduced amount of lesson time given to the
intervention and created tensions for tutors, teachers and students.
The algebra module uses a range of contextual situations and their associated models.
The use of models is more subtle than in number and more wide ranging, and students
were not always able to see how to relate the formal demands of a question to the
informal strategies developed during the sessions.
Some of the contexts used were very well received, including the ‘chip shop’ situation
where students were able to develop reasoning and representations relating to
simplifying, expanding brackets and factorising. The context of the seesaw and
traditional weighing scales helped students to become successful in solving equations
with ‘x’ on both sides. By representing a formal algebraic equation with an informal
weighing scales picture, students unable to answer this question in their pre-test were
able to deduce a solution. An example is shown in Figure 5-1.
The Algebra module delivery may have benefitted from prioritising the order of sessions
in order to build from the success of the Number module and in particular to focus
initially on developing use of the bar model for algebraic thinking.
5.2.3 Timing and student engagement
The modules for this intervention were designed to take into account the relatively short
amount of time available within a GCSE resit course, while balancing this with the
deliberately slow pace of an RME approach. In reality, the contracted version of the algebra
Figure 5-1 Algebra post-test example of a student using an informal strategy to solve an equation
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module and the length of time taken for students to open themselves to these approaches
would suggest that a longer intervention could have more impact. At times researchers
leading the lessons were forced to compromise RME principles in order to keep up to date
with the coverage contract established with the host institutions. Teachers working with
RME approaches for, say, the first term of a GCSE resit course would be able to devote more
time to following student thought processes and promoting group interactions.
The positioning of the modules within the GCSE resit courses may have led to other issues.
The number module was situated part way through the first term after host teachers had
already established their classroom norms, meaning that classes viewed the intervention
teachers as ‘different to usual’. For some students this caused a reluctance to engage, and
this seemed to be more evident when classes had been working with their own teachers for
longer. In an earlier pilot phase, the intervention had commenced at the start of the year to
good effect. The Algebra module fell towards the end of the second term, when the
pressures relating to speed of coverage of topics were most pronounced.
Students’ expectations of classroom norms relating to mathematics teaching and learning
meant that some found it difficult to engage with how they were being asked to think and
work. Setting up an RME culture where the student, not the teacher, makes decisions about
what strategies to use and whether a solution strategy is correct or not, takes time; with
two of the classes this was difficult to achieve within the confines of the short modules. This
investment of time has the potential to hugely benefit students and would be easier to
achieve within a longer intervention.
Once a formal procedure is taught to a class, there is often a shift of authority away from
informal strategies, and a pressure to use the standard procedure and related reluctance on
the part of the learner to use a method that they may now perceive to be inferior. This
attitude was certainly present amongst some of the students we worked with, who were
initially resistant to working with informal methods, perceiving them to be a backward step.
This raises the question of how to adapt the intervention to help convince students to value
the informal more. It also raises a much bigger question about the way mathematics is
taught to students who find themselves in GCSE resit classes and in particular whether, how
and when lower attaining students should be taught formal procedures.
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6 Implications and recommendations
Overall, our curriculum design objectives were to produce materials that were true to the
principles of RME but covered a significant amount of the Foundation Tier GCSE curriculum.
However, while we have seen some success, particularly in terms of the qualitative analysis
of the Number module tests, the speed of introduction of these models and the accelerated
process of ‘progressive formalisation’ of the models may have impacted on the success of
the intervention in terms of impacting on students’ performance in examination conditions.
In the Netherlands, RME is the adopted pedagogy from the moment that children begin
their formal education and the approach is sustained throughout their schooling. In the
Post-16 sector, RME will be ‘competing’ with other known but often misremembered and
misunderstood strategies. Within this context, a number of implications arise from our
findings with respect to mathematics teaching and CPD.
6.1.1 Recommendations for mathematics teaching at GCSE resit
6.1.1.1 The benefits of a two year course
We recommend that a two-year course would be beneficial, particularly for students who
have achieved well below a standard pass grade. There are many students who repeat GCSE
twice and achieve the same grade, or even worse, both times. Very few students are likely
to move securely to a standard pass in what is at most a 9 month course. As we have seen,
many lack basic skills. A two-year course would give teachers the time to not only address
basic skills gaps, but to interest students in mathematics, explore the often deep-seated
misconceptions that students have at this level, promote genuine understanding, and
produce students who not only have a better grade in mathematics, but also a greater
facility to use the subject in the future. The importance of such a shift is underlined by the
changes in the new 9-1 specification towards greater depth, more focus on problem-solving,
and the provision of clear mathematical arguments.
6.1.1.2 Incorporating a full term 1 RME approach in one year courses
We recognise the pressures on providers to offer resit courses as one year courses only. In
this context, we recommend that the benefits of the short intervention in this project are
extended and consolidated by employing RME-based materials and pedagogy throughout
term 1. This would enable coverage of more content and work on unifying models which
would provide a robust base for a more effective traditional focus on revision and past
papers in terms 2 and 3.
6.1.1.3 Building student learning skills
Students not only need to improve their basic skills: they need to develop learning skills.
Many students did not believe that mathematics made sense, and consequently never
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reflected on their answers. They needed to develop skills of questioning their own and
others’ strategies, and of sharing and evaluating ideas. Interaction is a key principle of RME
and needs to be fostered and to some degree learnt by students used to traditional
teaching. We recommend a 2-week introductory module focussing on learning, with regular
re-visiting throughout the course. Given the current pressure on time, this would ideally be
part of a two-year course, although this could be time well spent even in a one-year course.
6.1.1.4 Building a classroom culture that supports learning
Post-16 resit students understandably feel that they have ‘failed’ in mathematics. They have
little confidence in their own ability, and just want ‘to be told ‘what to do. The analyses in
Section 4 show students who are reluctant to engage in genuine discussion about
mathematics or to make any real effort to ‘make sense’ of the subject. Yet these are crucial
elements if they are to improve their understanding and ultimately their attainment. Hence,
teachers need to effect a shift in classroom culture and student attitude. Section 4 clearly
shows that this is possible, but it requires skill and intent on the part of the teacher and also,
again, it requires time. It is also important that teachers have materials, classroom activities,
and ways of working which support this shift, and RME provides this. Part of the problem
with the notion of a ‘resit’ class is that it hints at the fact that only the exam counts and the
only success criteria is a ‘pass’. In this context, neither teachers nor students are really
encouraged to explore the subject, create interest, or deepen understanding. This project
has shown the potential for RME to do these things while still improving student
performance.
6.1.2 Implications for Continued Professional Development (CPD)
The independent evaluation report recognises the need for “effective and achievable CPD
for teachers” and asks how this might be constituted. The number of teachers who have
already signed up for the CPD courses which we have developed as a result of this project is
an indication of how much they feel the need for support at this level. There is recognition
that ‘doing the same thing over again’ is not just unproductive, but also demoralising for
both students and teachers. We sense a genuine desire for change. To effect change,
however, teachers need both materials that support the change, and also regular
Professional Development. A simplistic ‘front-loaded’ CPD model will not be sufficient: the
pressures on teachers, well documented throughout this report, will see them reverting to
previous practice. Section 4 clearly shows that using RME materials can bring about a
change in classroom culture in which students will engage in discussion and sense-making. It
also shows, however, that this is a difficult process and that teachers need support if they
are to effect this change.
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6.1.2.1 Lesson study as support for substantial and sustainable CPD
Many GCSE resit teachers’ knowledge of mathematics is likely to be procedurally
dominated, so that even if they want to operate in a different way, they may not have the
expertise to do so. RME can support the development of teachers’ pedagogic and subject
knowledge: models bridge the gap between informal and formal understanding, and their
use enables teachers to feel less pressure to move to early formalisation. A well-designed
RME-based CPD has the potential to develop teachers’ support for students working at
differing levels of abstraction, so that those who find more formal notions difficult can
continue to make progress and develop strategies for solving problems. We recommend
that CPD provides opportunities for teachers to meet regularly with curriculum designers
and other teachers in order to understand the relationship between the design principles
and the materials.
One way of providing a CPD framework to support this development is Lesson Study (see
Huang & Shimizu (2016); also http://tdtrust.org/what-is-lesson-study). This model involves
teachers working with RME materials and planning lessons in pairs, teaching lessons and
observing each other, and feeding back on course days where they have the opportunity to
analyse and reflect on their implementation of the material with other teachers and RME
tutors. We recommend that this model is followed through the course of the academic
year, with up to 5 course meetings, enabling teachers to develop their pedagogic and
subject knowledge in ways that will support a sustained RME approach in the challenging
environment of the GCSE resit classroom.
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8 Appendices
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8.1 Appendix 1 – The pilot study, 2012-2013
The majority of the pilot took place in a large further education college, working with one
teacher and two groups of students with approximately twenty in each group. There were
two interventions, in October (Number) and January (Algebra), both consisting of around 10
hours teaching together. Some lessons were delivered by the usual class teacher, some by
the research team, and some by both. The students varied in age from 17 to 50, and were
studying mathematics GCSE for a wide variety of reasons. We focus here on student
performance in Number.
Students were tested on standard GCSE Number questions prior to the intervention. One
week after the intervention finished, students were given the same GCSE questions as a
Post-test. Two months later they were given different GCSE questions of a comparable level
of difficulty to the original questions, on the same topic, for questions 1 and 2 only. Table 8-
1 shows the percentage of students who answered each question correctly.
Number: GCSE type
questions
Percentage of students answering correctly (n = 39)
Pre-test Post-test Delayed post-test
Q1 Rates 35% 82% 65%
Q2 Percentages 28% 72% 69%
Q3 Comparison 11% 68% N/A
Table 8-1 Pilot results
The data show improvement in the performance of the students on all three questions. Of
particular note are the delayed Post-test results for questions 1 and 2, which suggest that
for most students the improved performance extended beyond the short term.
Qualitative analysis of pre-test questions: many students left questions blank, or wrote
‘can’t do this’. Sometimes they wrote down an answer with no working or explanation.
Evidence of students attempting to ‘make sense’ of the problem was rare. Interviews
revealed that their strategy often was to attempt to remember previously taught
procedures which had not been understood, or had been committed to memory but then
forgotten.
Qualitative analysis of post-test questions: many students filled the page with their own
individual versions of the RME models experienced in the intervention phase. They were not
only successful at adopting these strategies, but also showed confidence and flexibility in
making their own choices about how to apply them to a range of topics in Number.
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Interview data: We interviewed a sample of students after they had completed the post-
test. The focus was on the strategies they had employed on both pre- and post-tests. For
example, one student had written very little prior to the intervention when answering the
GCSE questions. For two of the questions she simply wrote down answers (one correct, one
incorrect), for others she wrote nothing. By contrast, in the post-test, her solutions included
diagrams using models from the interventions, which showed clarity and structure. At
interview she was able to explain her reasoning and commented with confidence on how
she was now able to engage with a problem using the models as a basis.
Overall findings of the pilot study
Students showed sustained improvement in their performance in Number.
They were able to adopt the RME models of the bar and the ratio table and by the end
of the intervention could successfully apply these to solve a wide variety of problems in
Number.
Students showed evidence of adopting sense-making approaches to solving problems in
Algebra. This enabled them to make progress in solving equations, expanding brackets
and solving simultaneous equations, although several students appeared to need more
time to develop their understanding of the RME-based approach to Algebra.
The Algebra material needed to be re-designed to take account of the limited time
available. In particular the material needed to enable students to build on their success
of using the bar model in Number to solve a variety of problems in Algebra.
The visual nature of the RME models helped many students, and the idea that drawing a
picture to represent a problem was a tool for ‘unlocking’ students who felt they were
‘stuck’.
Teachers need support if they are to adapt an RME approach in the Post-16 context. In
particular, they need help to recognise how widely applicable the RME based methods
are to solving a large number of GCSE problems. They also need help to recognise that
RME methods do not necessarily need to replace the methods that students bring
themselves, but instead can be used to enable them to develop an understanding of
why their particular method works.
The pilot study demonstrated the potential benefits of RME with this group of students and
teachers. It also highlighted the need for further work on understanding the challenges for
this group, and the specific materials requirements.
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8.2 Appendix 2 – Design principles
Examples from the teaching materials illustrate the use of RME theory within our curriculum
design.
Figure 8-1 Introducing the context
Figure 8-1 introduces a context that involves Aidan investing his redundancy money in a
sweet shop. Although the context is not real to 16 year old students it is ‘realisable’. Van
den Heuvel-Panhuizen (2005) and Gravemeijer, Van den Heuvel & Streefland (1990) suggest
that a context is valid if it is ‘realisable’ to the student and that with younger students, fairy
tales often provide useful contexts. At this stage, the question posed to the group is non-
mathematical and is designed to get them involved in the context.
Figure 8-2 Introduction of fair sharing
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In Figure 8-2, the concept of fair sharing is introduced with a question which provides
purpose to a mathematical task. Providing a sense of purpose is a key element in the
materials.
Figure 8-3 Introduction of ‘model of’
As we move onto Figure 8-3, the marshmallow is represented as a long rectangle or bar. In
other words, the bar has become a ‘model of’ the marshmallow (Treffers, 1987). This ‘model
of’ provides an informal representation of the mathematics but remains closely connected
to reality (in this case, the marshmallow tube).
Figure 8-4 Repetition of ‘model of’
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A second context is introduced in Figure 8-4, again leading to a rectangular representation
that is close to reality. This again results in a ‘model of’, and it is the repeated use of ‘models
of’ that will allow students to appreciate the generalisable use of the bar model and thus
begin to use it as a ‘model for’. Fraction notation is introduced here, but within a problem
that is very close to reality.
Figure 8-5 Bridging the gap between informal and formal
A third context is considered in Figure 8-5, again encouraging the students to draw a bar.
However, here the bar represents a distance of 15km and students are required to find
specific fractions of that distance. The bar has become more abstract at this point, and some
students may represent it as a double number line. Van Den Heuvel-Panhuizen (2003)
describes this as bridging the gap between informal understanding overtly linked to reality,
and more formal mathematical systems. This gap is closed further in the next slide.
Figure 8-6 Closing the gap and the top of the ‘iceberg’
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Question 2 of the slide depicted in Figure 8-6 presents problems which are approaching the
abstract world of mathematics and would be at the top of Webb’s ‘iceberg’ (Webb et al.,
2008).
Figure 8-7 Use of models in multiple areas in mathematics
Figure 8-7 shows an activity from later in the series of lessons. Now the bar has become a
stacked bar chart, and at this point students will draw pie charts from the bar chart. This
exemplifies the repeated use of models in a variety of areas of mathematics thus making
effective use of the limited time available in Post-16 resit courses.
Figure 8-8 Visualising fractions to make sense of common denominator
In Figure 8-8, a segmented bar is being used for adding fractions. The purpose here is to
provide a model that will form a visualisation of the fraction and allow students to make
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sense of the use of common denominators. Students with a recollection of adding fractions,
however strong or weak, map their solutions onto the bar to allow them to gain further
insights into the mathematics.
Figure 8-9 Using the bar in a familiar context
The problem in Figure 8-9 is introduced in the early stages of a lesson on percentage. The
bar is again used in a context familiar to the students and they will be encouraged to discuss
the context.
Figure 8-10 The bar as a ‘model of’
As we progress through this unit of work, the students increase and reduce amounts by a
given percentage (see Figure 8-10). At this point, the bar has become a ‘model of’, with the
students using it for a variety of questions involving proportional reasoning (Treffers, 1987).
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Figure 8-11 Recognising the limitations of the bar
The bar model in Figure 8-11 is close to reality but here students are encouraged to become
aware of a limitation. They will understand that in certain situations there will be a need to
make conversions between much larger measurements but that the bar will not allow this
as it is a representation which is drawn to scale. This creates the need for a new
representation and a purpose for the introduction of the ratio table.
Figure 8-12 Ratio tables as a more flexible model – a ‘model for’
The ratio table in Figure 8-12 is a more flexible model as it allows proportional pairs to be
calculated and recorded without needing to be to scale. Soon after this, students will begin
to use the ratio table to record in non-ascending order to answer a variety of problems. The
ratio table is a ‘model for’, and could be considered to be a ‘pre-formal’ model. It does not
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resemble the context that it represents, and yet the use of real life units on the left hand
side allows the user to refer to the context. For example, doubling the numbers in a column
would make sense (twice the inches would equal twice the centimetres). However, adding
two to the top and bottom would not make sense (two more inches would not mean two
more centimetres). Middleton and Van den Heuvel-Panhuizen (1995) concluded that after
being exposed to ratio tables,
.. students who traditionally had trouble with computation began to perceive the
underlying structure of the mathematics and became more proficient at computation
of rational numbers
This would suggest that the ratio table would be a useful tool to develop for Post-16 resit
students who typically have poor computational skills and number sense. Along with the bar
model, the ratio table is an example of a pre-formal model that van den Heuvel-Panhuizen
describes as a means to bridge the gap between real life problems and formal mathematics
(Van den Heuvel-Panhuizen, 2003).
116
8.3 Appendix 3 – The number and algebra module contents
Content of Number module at a glance:
Unit
Content
The Sweet shop
(4 hours)
Using contexts which lead to bar model representations for fractions
Developing the idea of what a fraction is
Comparing fractions using various strategies
Drawing and using a bar model to find a fraction of an amount
Drawing and using the bar model to divide in a given ratio
The Canteen Survey
(4 hours)
Making connections between the bar model (segmented
strip) and the circle model (pie chart)
Developing ways to compare, add and subtract fractions
using a bar (segmented strip)
Division of fractions using a bar
Solving Problems
using the Bar Model
(3-4 hours)
Using contexts that lead to bar model representations for percentages
Drawing and using a bar model to find a percentage of an amount and percentage increases and decreases
Finding the original amount after a percentage change
Problems involving depreciation and repeated percentage change
Problems developing an understanding of proportional reasoning
Ratio tables
(4 hours)
Using ratio tables to solve problems associated with
proportional reasoning;
Direct proportion
Recipes
Conversions
‘Best buy’
Unitary method
117
Content of Algebra module at a glance:
Unit
Content
I Think of a Number
(1.5 hours)
Introducing algebraic notation.
Solving equations where the unknown appears once.
Rearranging equations where the unknown appears once.
Graphs
(1 hour)
Reading expressions
Creating tables of values and graphs of linear and quadratic functions
Fish and Chips
(1.5 hours)
Simplifying algebraic expressions
Expanding and factorising algebraic expressions
Easy to See
(1 hour)
Encouraging students to make sense of algebraic equations.
Solving a variety of equations, including quadratic, using common sense strategies.
Word problems using
a bar model
(2 hours)
Enabling students to represent a variety of word problems using one or more bars
Developing strategies for comparing bars in order to solve a variety of word problems
Recapping and developing the use of the bar as introduced in the Number module
Balance
(1.5 hours)
Using the context of weighing scales to develop a conceptual understanding of balance
Solving equations with ‘x’ on both sides and brackets, using the context of balance
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8.4 Appendix 4 - The survey bar lesson
Slides used in Lesson A and Lesson B
119
120
8.5 Appendix 5 – The independent evaluation report
Independent evaluation of ‘Investigating the impact of
Realistic Mathematics Education approach on
achievement and attitudes in Post-16 GCSE
mathematics resit classes'
Mark Boylan and Tim Jay
Sheffield Hallam University
121
8.5 Appendix 5 – The independent evaluation report ................................................ 120
8.5.1 Introduction ..................................................................................................... 123
8.5.2 Evaluation methodology and methods............................................................ 124
8.5.2.1 Evaluation approach and rationale .......................................................... 124
8.5.2.2 Measures of impact .................................................................................. 125
8.5.2.2.1 Attainment measures .......................................................................... 125
8.5.2.2.2 Attitude measures ................................................................................ 126
8.5.2.3 Analysis ..................................................................................................... 126
8.5.2.4 Process evaluation .................................................................................... 127
8.5.3 Test conduct and marking ............................................................................... 127
8.5.3.1 Conduct of the tests ................................................................................. 127
8.5.3.2 Test attendance ........................................................................................ 128
8.5.3.3 Test marking and moderation .................................................................. 128
8.5.4 Data analysis .................................................................................................... 129
8.5.4.1 Balance and attrition ................................................................................ 129
8.5.4.2 Did the intervention raise levels of attainment? ..................................... 129
8.5.4.3 Was there increased use of a RME approach in the intervention group?
131
8.5.4.4 Analysis of attitudes data ......................................................................... 133
8.5.5 Implementation evaluation findings ................................................................ 133
8.5.5.1 The RME GCSE resit intervention ............................................................. 133
8.5.5.1.1 Rationale and theoretical background................................................. 133
8.5.5.1.2 Recipients and contexts ....................................................................... 134
8.5.5.2 Implementation ........................................................................................ 135
8.5.5.2.1 The intervention ................................................................................... 135
122
8.5.5.2.2 Differences between the intervention and usual teaching approaches
135
8.5.5.2.3 Issues affecting intervention delivery .................................................. 136
8.5.5.3 Responses ................................................................................................. 137
8.5.5.3.1 Student responses ................................................................................ 137
8.5.5.3.2 Teacher views on impact on students and on delivery ....................... 138
8.5.5.4 Issues potentially affecting the security of the evaluation ...................... 138
8.5.6 The potential for future developments and trials ........................................... 139
8.5.6.1 Intervention design................................................................................... 139
8.5.6.1.1 Whole class intervention ..................................................................... 139
8.5.6.1.2 Developing a one to one intervention ................................................. 140
8.5.6.1.3 Professional development needs ......................................................... 140
8.5.6.2 The prospects for trialling future interventions ....................................... 141
8.5.6.2.1 Whole class intervention ..................................................................... 141
8.5.6.2.2 One to one interventions ..................................................................... 142
8.5.6.2.3 Implementation issues ......................................................................... 142
8.5.7 Conclusion ........................................................................................................ 142
8.5.7.1 Limitations ................................................................................................ 142
8.5.7.2 Conclusions ............................................................................................... 144
123
8.5.1 Introduction
Sheffield Hallam University (SHU) undertook an independent evaluation of the Nuffield
Foundation-funded Manchester Metropolitan University (hereafter abbreviated to MMU)
investigation of the impact of Realistic Mathematics Education approach on achievement
and attitudes in Post-16 GCSE mathematics resit classes GCSE resit intervention. For brevity
the intervention is described in the report as the RME GCSE resit project.
This report should be read alongside MMU's description of their intervention in Sections 1-3
of the main report, which provide further detail of the intervention, its theoretical
background, materials and activities. A summary is provided here as part of the
implementation evaluation.
In the main report, previous implementation and research on the Realistic Mathematics
Education (RME) intervention is described. In the post-16 context a recent review of
interventions that have the potential to support attainment at a GCSE equivalent has been
undertaken1 l. This considered both previous use of RME and also an intervention that ha
some similar features - for example, the use of mathematical models and an emphasis on
conceptual understanding2. This review suggests that there is some evidence that
interventions similar to RME could have a positive effect on attainment. Given that, the
Post- 16 GCSE resit intervention is a newer context for RME. Therefore, MMU undertook a
relatively limited intervention in terms of number of students involved and the number of
hours of intervention across the academic year, to test RME in the GCSE resit context and to
research the use of an RME approach in this context in terms of students’ and teachers'
experience, and practical issues of implementation.
In addition, the intervention focused on two areas of mathematics - aspects of number
(proportional reasoning) and elements of algebra.
In the intervention, the researchers worked directly with the students, and class teachers
had the opportunity to observe. Whilst MMU had undertaken development work in GCSE
resit contexts and so materials had been piloted, in terms of a trial methodology this could
be considered a pilot3.
1 Maughan et al. (2016). Improving Level 2 English and maths outcomes for 16 to 18 year olds Literature
review. London: EEF.
2 Swan, M. (2006). Learning GCSE mathematics through discussion: what are the effects on students? Journal
of Further and Higher Education, vol. 30, no. 3, 229—241.
3 See
https://v1.educationendowmentfoundation.org.uk/uploads/pdf/Classifying_the_security_of_EEF_findings_FINAL.pdf
124
The aims of the evaluation were:
(i) To provide an independent evaluation of the impact of the RME approach as
operationalised in the project, on Post-16 GCSE resit students' achievements and attitudes
in mathematics.
(ii) To provide independent advice to the MMU team on issues of fidelity and teacher CPD.
(iii) To provide independent advice on the scalability of the intervention for a larger efficacy
trial using a randomised controlled trial methodology.
8.5.2 Evaluation methodology and methods
8.5.2.1 Evaluation approach and rationale
A quasi-experimental design was appropriate for this pilot in which 4 classes in three
different locations received the RME intervention and 4 classes in the same locations did
not. The quasi-experimental design allowed:
comparison of outcomes for students in relation to attainment and attitudes
comparison of student learning outcomes as a result of a limited experience of
alternative RME teaching as a supplement to prevailing teaching in post-16 and usual
approaches
identification of professional learning implications
However, there are limits to the reliability of the comparison of impact given the size of the
samples and the fact that this was necessarily a clustered trial. Further, constraints on
recruitment of the intervention and comparison groups meant that randomisation at class
level was not possible4. MMU were able to recruit teachers who were willing to have their
classes participate in the project and so these were assigned to the intervention condition.
Given this, there is a threat to the security of the trial due to possible imbalance between
the intervention and comparator samples on both observable and likely non-observable
relevant characteristics.
We compared impact between intervention and comparison classes on areas of
mathematics targeted by the intervention - aspects of number and algebra. There were
three phases to the evaluation of impact, corresponding to the two phases of the
intervention plus a final delayed post-test.
4 For this reason, the trial did not meet the standards for CONSORT registration.
125
The testing approach was designed to minimise the disruption for students and classes
involved, an important ethical consideration and one designed to help to reduce attrition.
The delayed post-test included a mixture of number algebra and other GCSE mathematics
questions. The delayed post-test aimed to assess whether there was evidence a sustained
impact of the intervention. However, it is important to note that the length of the delay
varied between number and algebra module teaching and the delayed post-test.
8.5.2.2 Measures of impact
8.5.2.2.1 Attainment measures
MMU and CDARE agreed the overall test design, with questions/sections populated by
MMU. Questions were based on or adapted from GCSE questions thus leading to a degree
of external validity of the assessment tool.
The Phase 1 test, focused on number, used the same questions in the pre-and post- tests.
This was to support planned diagnostic interviewing. The test had 7 questions, of which 2
asked for a self-assessment of whether the answer was correct and for an explanation of
this. This generated qualitative data about learners' relationships to mathematics and their
strategies.
The Phase 2 test contained 7 questions focused on algebra and used the same questions in
the pre-and post- tests.
The delayed post-test contained 15 questions, 5 on number, 5 on algebra and 5 on other
topics.
Tests were marked by MMU researchers using agreed mark schemes, based on a GCSE
approach giving marks for methods as well as final answer.
In addition, papers were analysed using the following criteria:
Attempted/not attempted (1-0) – to provide a data source on possible impact on
resilience.
Applies a relevant RME model appropriately (1-0) – to assess the effectiveness of RME
teaching regardless of effect of impact. This also provided indirect evidence of whether
the RME intervention had increased the students’ range of appropriate methods – for
example a student might have answered a question correctly in the pre-test and in the
post-test answered the question correctly again, but using a different method.
If using an RME approach, makes sense of the problem (1-0) – whether an RME
approach was used in a way that was appropriate and/or potentially productive to
provide data on impact on sense making.
126
8.5.2.2.2 Attitude measures
The attitude scale was developed by MMU and was adapted from the Understanding Post-
16 Participation in Mathematics and Physics Project5.
Additional items were designed to identify epistemologies of mathematics in the target
group in the RME GCSE resit project.
8.5.2.3 Analysis
Statistics were generated regarding attrition and balance across conditions.
The attitude scale was analysed by Principle Component Analysis by SHU. Reliability analysis
(Cronbach's alpha) was used on the resulting factor in order to test for internal reliability. It
was outside of the scope of this evaluation to assess test-retest reliability.
Analysis was undertaken to consider if attitudes to mathematics are affected by
participation in the RME intervention. This analysis was an ANCOVA, with 'attitude to
mathematics' as the dependent variable, 'experimental group: intervention/control' as
independent variable, and controlling for attitudes to mathematics at pre-test.
The analysis also considered if attainment in mathematics was affected by participation in
the RME intervention. Alongside the analysis of attainment, we carried out a similar
analysis for the dependent variable 'use of RME principles' as described above in order to
determine whether students in the RME intervention group respond to questions in the
target and non-target domains using RME approaches (e.g. visualisation and modelling). The
measure of difference in attainment could be used to calculate a Minimum Determinable
Effect size (MDES) for future studies.
The analysis considered if there are relationships between attitudes and attainment. This
analysis helped to assess associations among the variables measured. Part of the rationale
for the intervention is that an RME approach can lead to different perceptions of
mathematics and of how to go about doing mathematics, which will in turn lead to higher
levels of attainment. Multiple regression analysis was used to assess relationships among
attitudinal factors, use of RME principles, and effects on attainment in both target and non-
target domains.
5 Reiss, M, (2012) Understanding Participation in Post-16 Mathematics and Physics (UPMAP) ESRC End of
Award Report, RES-179-25-0013. Swindon: ESRC. http://research.ioe.ac.uk/portal/en/projects/understanding-
participation-rates-in-post16-mathematics-and-physics-upmap%2845a02dd7-1141-4e70-a97b-
bcb65adc79f4%29.html
127
8.5.2.4 Process evaluation
SHU undertook a light touch process evaluation, consisting of the following activities.
Visit to 1 college during phase 2 and write up. The visit included a joint observation with
an MMU researcher to evaluate the use of the observation protocols, and to consider
the RME approach in the classroom in relation to a possible efficacy trial.
Telephone interviews with the 4 intervention teachers (originally it had been planned
that the college visit would include a face to face interview with the host teacher, but
this was not practical).
Review of MMU research processes in relation to both intervention and control teacher
practices.
8.5.3 Test conduct and marking
8.5.3.1 Conduct of the tests
There were no issues reported with the conduct of the number test.
There were a number of issues that the MMU team identified in relation to the algebra test.
Test conditions for the post-test were not consistent. Due to the distance involved and
time constraints, at one site intervention and control students were supervised by their
class teachers for the post-test. During post-test interviews two students from the
control groups at the same site revealed that their teacher had helped them to answer
questions during the test. Closer examination of scripts from this site revealed that some
control students produced very similar responses to particular questions. At the other
two sites supervision was by one of the intervention staff.
One control group is known to contain more able students than the corresponding
intervention group due to the way the students are grouped.
RME marking. It is much harder to see overt use of RME methods in the case of Algebra
than number. In the case of the Algebra test, it was only possible to see use of RME in
some of the questions. It is possible that students were using RME models to answer
questions (but this could only be revealed through in depth post-test interview).
There were also issues reported by MMU with the delayed post-test:
Test conditions for the delayed Post- test were not consistent. At one site students did
half the questions as part of a GCSE mock paper. They did the other questions later. This
led to a loss of data as several students were absent (in particular, those in the
intervention group were not present for the second half of the test).
Researchers supervised the tests for three pairs of intervention and control classes. At
the other site, the same one where issues were identified for the algebra test, class
teachers supervised. At this site, there was evidence that some students were allowed
128
access to calculators and that in some cases students had been given help by their
teacher.
8.5.3.2 Test attendance
MMU undertook an analysis of patterns in test attendance. The results show a pattern of
increasing absence from courses. Although this information is based on just 5 data points, it
is worth pointing out that students did not always know that they were going to take a test,
and so each point represents a normal day as far as they were concerned. Absence rates are
very similar for intervention and control groups, with better attendance by intervention
students with one exception – the algebra post-test. However, this greater absence was not
statistically significant. There was one incident of a significant difference between the
groups: absence from the algebra pre-test was far lower for the intervention students (chi
square = 7.600, df = 1, p = 0.006). Percentages are listed in the table below, in order of
occurrence through the year.
Test (* p=0.006) Intervention group
absence
Control group absence
Number pre-test 12% 15.3%
Number post-test 20.3% 23.6%
Algebra pre-test 12% 30.6%*
Algebra post-test 37.8% 30.6%
Delayed post-test 31.1% 41.7%
Table 8-2 Test absence
8.5.3.3 Test marking and moderation
MMU blind marked the tests. Tests were mixed for intervention and comparison students
clustered in each school and then double marked by the researchers who were not directly
researching/teaching in that school. Following moderation MMU derived an agreed mark.
MMU provided a data file that included details of student demographics and other key data
(gender, age, prior GCSE grade, when GCSE last taken, college, class teacher).
Due to the reasons identified above concerning the conduct of the algebra test and initial
analysis of outcomes (see below) for efficiency, SHU undertook moderation of number tests
only.
SHU undertook a process of randomisation to identify a sample of the number test that was
then requested from MMU. The stratification matrix sampled by groups and between
intervention and control. The sample was moderated to evaluate consistency, and following
129
blind marking by SHU a 100% measure of inter-rater agreement was found between SHU's
marking and MMUs.
8.5.4 Data analysis
This section reports findings from the analysis of data collected by the MMU project team.
The first part of the section focuses on addressing the question of whether the intervention
had an effect on students’ attainment in number or algebra. The second part addresses
some further questions about outcomes of the intervention, including those focusing on
students’ use of RME approaches in their solution strategies and on potential group
differences in response to the intervention. The third and final part of this section focuses
on questions relating to the attitude survey completed by students before and after the
intervention.
8.5.4.1 Balance and attrition
Data were collected from 147 students, in eight groups. 75 students, in four groups, were
part of the intervention group, while the remaining 72 students, in another four groups,
were part of the control group.
Missing data were an issue for all analyses in this section. Complete data (pre-, post- and
delayed-post-test scores for both number and algebra) were received from only 52 students
(29 intervention, 23 control). Numbers will be reported for each analysis, and students are
included in each analysis for which complete data were collected; i.e. a student will be
included in analyses of number data even where there may be algebra data missing. The
high level of attrition has implications for interpretation of findings, although the post-16
mathematics retake context is often subject to similar rates of attrition.
8.5.4.2 Did the intervention raise levels of attainment?
This section essentially aims to answer four questions. Did the intervention lead to higher
levels of attainment in number and/or algebra for those in the intervention group, relative
to the control group, at either post-test, or delayed post-test?
Was there an effect on attainment in number at post-test?
Was there an effect on attainment in algebra at post-test?
Was there an effect on attainment in number at delayed post-test?
Was there an effect on attainment in algebra at delayed post-test?
Figure 8-13 shows pre-, post-, and delayed test scores for number. This shows that scores
for both groups increased. The intervention group started at a lower level, but at post-test
and delayed test performed at close to the same level as the control group. Note that
delayed test scores reflect results from a test comprising number and algebra items which
were testing the same concepts as the pre- and post-tests, but with different questions; the
130
delayed post-test also had a higher maximum score, and so these scores are not directly
comparable with pre-and post-test scores.
Figure 8-13 Effect of intervention on attainment in number
An ANCOVA with group as independent variable (44 students in the intervention group, 52
in the control group), post-test number score as dependent variable, and pre-test number
score as covariate shows a significant effect of group (F1,93=4.55, p=0.035). We can interpret
this as evidence that the intervention has been effective, although the effect size is small:
partial eta squared =.047, meaning that 4.7% of the variance in post-test scores can be
accounted for by participation in the intervention (Cohen's d = 0.26, based on adjusted
means). Much of the effect appears to consist of the intervention group ‘catching up’ with
the control group. The groups were not balanced in terms of prior attainment, and without
further investigation it is not possible to tell whether the observed effect may be due to
regression to the mean, different test conditions for the two groups, or some other factor.
An independent t-test looking at the difference in score on the number post-test between
the two groups did not reveal a significant effect (t=1.51, df=109, p=.134). However, we can
conclude that, for the Number assessment, the intervention group improved to a greater
extent than did the control group, between pre-test and post-test.
A second ANCOVA was carried out, with delayed test score as the dependent variable. This
shows no effect of group (F1,71=0.229, p=.634).
131
Figure 8-14 shows pre-, post-, and delayed test scores for algebra. It appears to show that
the intervention group performed at a lower level at both pre- and post-test, but at a higher
level in the delayed test, relative to the control group. Again, delayed test scores are not
directly comparable with pre- and post-test scores as they derive from a different test, with
a higher maximum score.
An ANCOVA with group as the independent variable, post-test algebra score as dependent
variable and pre-test algebra score as covariate showed no significant effect of group
(F1,75=1.08, p=.302). A similar ANCOVA with delayed test for algebra as the dependent
variable showed no effect of group (F1,67=2.61, p=0.11).
Figure 8-14 Effect of intervention on attainment in algebra
The only significant effect in this section is the greater improvement between pre- and post-
test for Number, by the intervention group compared to the control group.
8.5.4.3 Was there increased use of a RME approach in the intervention group?
Analyses from this point on in this section have been carried out only for data from the
Number tests. This is partly because of the above finding, that there was an effect of the
intervention on attainment in Number, but not Algebra, and partly because of reported
irregularities in the way that Algebra tests were carried out.
In this case, 'an RME approach' was observed through use of the bar model or the ratio
table.
132
Table 8-3 shows that, while neither group used an RME approach to answer questions
during the pre-test, the intervention group did use an RME approach to answer some
questions during the post-test. This difference between groups was significant (t=8.73,
df=97, p<.0005). Of the 49 students in the intervention group who took the Number post-
test, 36 used an RME approach at least once.
There is also a significant correlation between students’ improvement in score between pre-
and post-test in Number, and their degree of use of an RME approach (r=.258, n=86,
p=.016). However, when including only those students in the intervention group in the
analysis, the correlation is not significant (r=.227, n=44, p=.139), so this finding should be
interpreted with some caution.
Intervention N Mean Std. Deviation Std. Error Mean
Pre-test RME use control 59 .02 .130 .017
Intervention 64 .00 .000 .000
Post-test RME use control 50 .00 .000 .000
Intervention 49 2.76 2.232 .319
Table 8-3 Descriptive statistics for use of RME approach for number items
To find out whether increased use of RME could account for increases in performance in the
number test by the intervention group, a multiple linear regression analysis was carried out,
with post-test number scores as the outcome variable. Model 1 has just pre-test number
score as a predictor. Model 2 adds post-test RME frequency, and then Model 3 adds the
group variable indicating whether participants were in the intervention or the control group.
Table 8-4 shows that, after having taken account of variance due to initial variation in pre-
test scores, an additional 2.4% of the variance in post-test score can be accounted for by
variance in use of RME approaches (F1,83=6.083, p=.016). After both pre-test score and RME
use have been accounted for, then no additional variance is explained by whether the
participants were in the intervention group or the control group (F1,82=0.219, p=.641).
Predictors R R Square Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change F Change df1 df2
Sig. F Change
Pre-test only .807a .651 .647 2.722 .651 156.982 1 84 .000
Pre-test, RME use .822b .675 .667 2.643 .024 6.083 1 83 .016
133
8.5.4.4 Analysis of attitudes data
KMO (0.791) and Bartlett’s test of sphericity (p<0.0005) were used with the pre-test attitude
data to show that the data were suitable for factor analysis. A Principal Component Analysis
(PCA) revealed that a 1-factor solution gave the best account of the data, accounting for
31% of the variance.
A regression method was used to create a single score for ‘pre-test maths attitude’. The PCA
was repeated for the post-test questionnaire data, which included an additional 4 items. A
1-factor solution still explained the data best, so the additional 4 items were not included in
the single score for ‘post-test maths attitude’. The pre- and post-test attitude scores were
analysed for any differences by group or as a result of the intervention. A 2-way ANOVA
with group (27 students in the intervention group, 24 in the control group) and time (pre-
test, post-test) as independent variables, and attitude as dependent variable revealed no
significant main effect of group (F1,49=0.697, p=.408), no significant main effect of time
(F1,49=1.375, p=.247), and no interaction between group and time (F1,49=1.920, p=.172).
8.5.5 Implementation evaluation findings
8.5.5.1 The RME GCSE resit intervention
Here the intervention is briefly described. This should be read alongside the research report
and other publications by MMU which provide a fuller description.
8.5.5.1.1 Rationale and theoretical background
The intervention aimed to address the needs of 16-18 years old students and others who do
not obtain a grade C GCSE during compulsory schooling and for whom there is a low
conversion rate when resitting6.
The intervention is based on the Realistic Mathematics Education approach. Important
features of RME are:
6 Department for Education (2013) Level 1 and 2 attainment in English and Mathematics by 16-18 students.
Statistical first release accessed at www.gov.uk
Pre-test, RME use, group
.822c .676 .664 2.656 .001 .219 1 82 .641
Table 8-4 Linear regression statistics to show relationship between use of RME and post-test attainment in number
134
use of carefully chosen contexts that are meaningful to students and models to support
student visualisation and understanding of mathematical processes and concepts (a
meaningful context is one that can be imagined or visualised)
unlike other context rich pedagogies, context in RME is used not primarily for application
of mathematical learning but to support learning and conceptualisation
careful choice of models to bridge informal understanding to formal understanding and
abstraction and to connect different areas of mathematics and consequently a slower
movement towards formalisation
the encouragement of an explorative and problem solving approach by students
In the RME GCSE resit intervention the most prevalent models used were bar models and
ratio tables, but other models – for instance balance scales – were also used.
8.5.5.1.2 Recipients and contexts
The intervention took place in three sites involving four intervention host teachers:
an 11-18 single gender school (one intervention class, one control)
two large FE colleges (three intervention classes - two 16-18 years old classes, and one
adult/mature student class, three controls)
o Two of the teachers whose classes were involved in the intervention had
previously hosted MMU researchers developing and trialling materials for use
with the GCSE resit classes. One of these teachers described having adopted
some of the modelling approaches as part of their general practice as a result. A
third teacher worked at a college where MMU had previously researched but
had not personally been involved. There had not been any previous activity
related to RME by MMU on the fourth site.
As stated above the total numbers of students for whom evaluation data was obtained
were 75 in the intervention groups and 73 in the control.
Mathematics teaching in the settings for GCSE resit classes took place in lessons that
ranged from 1-3 hours long.
In the FE settings in particular teachers reported very challenging circumstances in terms
of staffing of GCSE resits, timetabling and student motivation, some of the issues
identified were
o the current practice is for GCSE resit course to last one year (and given the
examination timings less than a year's teaching)
o a prevalent culture in GCSE resit classes of focusing on examination with the
attendant danger of 'teaching to the test' and rapid progression through the
curriculum
o lack of resources or time for teacher CPD
o a culture of low priority or value of GCSE resits and a history of low conversion
rates arguably leading to low expectations
135
o irregular attendance of some students
8.5.5.2 Implementation
8.5.5.2.1 The intervention
MMU tutors taught the material with host teachers observing or taking some non-active
role in the lesson (e.g. marking at the back of the class). In some cases teachers attempted
to help individual students in lessons, although steps were taken wherever possible to
prevent this. For the most part, tutors would be elsewhere or be undertaking other tasks
whilst in the room.
MMU tutors used materials designed specifically for the project including presentation
materials and student activities on paper.
The number module was designed to last 12 hours of teaching and the algebra module 9
hours. MMU tutors taught RME during usual timetabled teaching hours, thus it replaced
usual teaching rather than supplementing it.
From information from telephone interviews and from the MMU team, the teaching time
was achieved for the number module, and apparently exceeded at the 11-18 school where
students had four hours of mathematics per week in one hour lessons. On one site the
timing of the algebra module was later than had been intended and there was disruption to
algebra teaching so the intended 9 hours was not fully achieved. Teaching time for the
Algebra module was limited partly due to the anxiety of teachers about coverage of the
course during the Spring term. While the module was designed to take 9 hours, only one
class received this amount, while two classes received 7.5 hours tuition and another 4.5
hours. This led to inconsistencies in which topics were taught with which classes. For
example, the bar for word problems was only taught to PT1; the balance scales for
equations to PT4.
From telephone interviews it appears the teaching approach was consistent by different
MMU tutors, with one significant adaptation/variation at one site being that lessons were
shorter but more frequent.
A feature of the RME approach is to work with contexts that are meaningful and to respond
to students' developing formulation. Therefore, exact replication of teaching to all classes
would be inconsistent within the theoretical framework. It was clear during meetings with
the MMU team that all team members had a consistent and shared view of the theoretical
framework and of RME. Thus, a relatively high level of implementation fidelity in terms of
quality can be inferred.
8.5.5.2.2 Differences between the intervention and usual teaching approaches
During the telephone interviews, teachers commented on differences between the RME
intervention and usual teaching approaches.
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All four teachers commented on the issue of pace of curriculum coverage, contrasting the
amount of time they have to teach a topic when trying to cover the whole examination
syllabus in approximately 90 hours of teaching time, with the amount of time taken to cover
material in the RME lessons.
Two discussed the usual approaches as being focused on the use of instrumentalist and
procedure orientated methods:
"here we give a technique to deal with each type of problem, we try drill them with a
way of doing a problem" (Teacher 2)
"we focus on a single method" (Teacher 3).
This was contrasted with the RME explorative approach, the development of understanding
and the use of model that is applicable to more than one type of problem.
A third teacher who had previously worked with MMU researchers has adopted some of the
approaches, specifically mentioning ratio tables. The fourth teacher, though new to working
with MMU described their teaching approach as similar to that of the MMU researcher,
though clearly there is no way of verifying this claim.
The teachers contrasted the intervention with their own usual teaching. A limitation of the
evaluation is that data was not collected on teaching approaches in the control classes.
8.5.5.2.3 Issues affecting intervention delivery
The MMU tutors found it challenging to negotiate time for teaching. Whilst when
interviewed teachers were enthusiastic about the approach and said they would use bar
modelling, they also commented on the need to move through the curriculum quickly. The
latter need was related to the one year length of the resit course and the limited time for
mathematics teaching.
In the FE settings variable attendance was an important issue identified by teachers as
affecting outcomes during the telephone interviews. One teacher described attendance as
generally being 60% to 80%. In the view of the teachers RME lessons were neither better
nor more poorly attended. It is important to note that some of the students in the trial are
likely to have experienced less than the 12 hours intended RME mathematics in number or
the 6-9 hours available for algebra. The issue of attendance amplified concerns by teachers
about curriculum coverage, given that they expected students to miss a number of lessons.
Resit students often have a negative relationship to mathematics, and many have prior
attainment significantly lower than the C grade they are aiming for. For many, GCSE 'failure'
is only the most recent in a long history of 'failing' in mathematics.
All four teachers commented that some, or in the case of the 11-18 school most, students
were resistant, in general, to new approaches to learning mathematics. For some this
appeared to be related to a lack of confidence. The two teachers in the FE colleges of the
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16-18 classes both noted how students who had attained a D in GCSE often had a view that
they only needed to gain a few more marks to convert their grade to a C. This was one
reason, it was believed, for absences in some cases. The teachers pointed to the way in
which students had been coached and drilled for GCSE at 16 and so often had significant
gaps in understanding.
8.5.5.3 Responses
8.5.5.3.1 Student responses
Student attitudes were inferred from teacher reports and the one observation. Students
were actively engaged in the RME lesson that was observed. Teachers reported that
students have a mix of responses to the RME lessons and materials: some were highly
engaged, other students rejected being taught new methods or approaches, focusing
instead on being taught 'how to pass the exam' and to use methods they had previously
been taught. One teacher stated that the RME approach was challenging for students as
they had usually been taught to "go straight for an answer" (Teacher 2).
Three of the four teachers related student disinclination to using new methods to students'
prior attainment or student self-perception of their mathematical ability. Students
described as relatively stronger were less favourable, initially, at least to the RME approach,
although one teacher commented on how this changed when the models were used for
harder problems that such students had not been able to previously engage with.
These three teachers contrasted this less favourable response amongst such students with
the response of the students with lower previous grades who were more receptive from the
outset. One commented "if you keep repeating the same ways then they just freeze up"
(Teacher one) and therefore RME was valuable as it offered a fresh approach. One teacher
in contrast suggested that it was relatively higher attaining students who could engage more
fully with the RME models.
One teacher also linked attendance to student response, as students who missed a session
then did not always find it easy to re-engage with the teaching. This contrasted with the
usual approach in which a topic or two was covered in a single session before moving on
and so lessons were self-contained.
However, from the interview data it is difficult to separate possible responses to RME
teaching from the student responses to new and different tutors. All teachers commented
on this issue, noting that it took some time for students to be 'won over', even though they
also commented positively on the MMU teachers’ capacity to build relationships with
students quickly. One of the teachers who had worked with MMU previously contrasted this
year with a previous year when the MMU teacher was able to meet the students at the start
of term and so relationships were built as part of the induction to the resit course. The
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teachers pointed to students' lack of confidence as a reason why they had some reluctance
to engage from the outset with new and different teachers.
This issue of initial student response may have lessened the effectiveness of the first session
of the intervention. Based on this, students in the intervention groups potentially only
experienced 9-10.5 hours rather than 12 hours of number RME teaching, and a similar
situation for the algebra module, which as described above already had less teaching time
than intended.
However, the overall view of the teachers was that students responded positively to the
RME lessons. Two teachers commented on seeing some students using the bar model in
later work separately and independently from the RME intervention lessons or tests.
8.5.5.3.2 Teacher views on impact on students and on delivery
All the teachers interviewed were positive about the RME approach, one describing it as
'brilliant' (Teacher one). As stated above, one had already tried to integrate some of the
models into her own teaching. The teacher in the 11-18 school believed the models were
valuable and would be more beneficial if introduced earlier to the students and so the
department was considering adopting/exploring RME with Y7.
Three teachers commented on the value of the models in helping students to problem
solve. This was both identified as a particular weakness for the resit students but also an
important issue given new GCSE specifications.
Two teachers pointed to the importance of embedding the approach in usual teaching. One
noted that after the intervention lessons both the students and himself quickly reverted
back to 'same old ways' (Teacher 2). He suggested a better approach would be for the
intervention to happen at the very start of the year and then for the models to be used
consistently.
Two teachers pointed to the need for appropriate professional development to support
them to use the RME approach themselves, but also pointed to the difficulties for FE
teachers to access professional development of any type. One teacher noted in particular
they would not have been able to apply the models to algebra themselves without
observing the MMU researcher.
8.5.5.4 Issues potentially affecting the security of the evaluation
It is important to note the possibility that some of the research aspects of the intervention
may have also had an effect on outcomes. In particular, one of the research team worked
with individual students in a small number of lessons using a data capture pen. Whilst such
conversations were not aimed at 'teaching' the students there is a potential effect on
attainment. In addition, as stated above, there was some variability in the extent to which
teachers observed lessons, or acted as participants in lessons supporting students, though
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steps were taken to minimise possible effects of this. Above, issues were noted that affect
reliability of the algebra test.
8.5.6 The potential for future developments and trials
8.5.6.1 Intervention design
This section draws on the data analysis findings, the implementation evaluation and findings
reported by the MMU team related to student engagement and learning and teacher views
and responses. These indicate that the intervention would benefit from further
development in order to increase the likelihood of having impact. Any further development
will need to address the issues identified above that effect GCSE resit classes, in particular
designing an intervention that takes into account attendance issues and the time needed for
unconfident students to adapt to the new approach. At the same time, the intervention
findings also underline the need to address the low GCSE conversion rate.
The outcomes of the test analysis and other data potentially could inform the MMU teams
reflection on the posited underlying change mechanisms. We suggest two possible
approaches that could be considered in relation to this: extending the RME intervention for
whole class teaching or alternatively developing a one to one intervention. Either approach
would require further development of the project as a professional and curriculum
development programme.
8.5.6.1.1 Whole class intervention
The evaluation findings suggest that one approach to developing the intervention would be
for it to extend over two years. This is because of the challenge of implementing the
intervention in the context of a single year course as well as the MMU findings and the RME
conceptual framework about the need to take time to develop understanding. Such an
approach would also address the need that can be inferred by teachers that students may
need to 'unlearn' both approaches and mindsets that are barriers to their success.
The number module had a short term small effect. Further, this effect was linked to the use
of the MMU RME approaches used in this module. The implementation evaluation suggests
that following the intervention there was some reversion back to previous techniques in at
least one class and some students. Issues of attendance were identified again potentially
lessening the impact of the intervention for some.
The positive outcome might be increased by extending this module and approach over two
years to address these issues.
In any case, any extended intervention would require attention to CPD for teachers so that
students would experience the intervention pedagogy for substantially longer than the 12
hours aimed at in the evaluated study. One teacher pointed to the value of such a module
being introduced early in the curriculum, possibly at the start of the year.
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The lack of observed effect of the algebra module may be due to challenges in
implementation. However, the analysis above suggests that the RME approach to algebra
for GCSE students may need to be further developed, in particular also requiring a more
sustained intervention.
If other aspects of the curriculum were to be addressed through an RME approach then the
evaluation findings also suggest that a longer and more sustained intervention would be
advisable. Notwithstanding that the evaluation provides no evidence of what impact RME
might have in other areas of the curriculum as these were not included in the intervention.
8.5.6.1.2 Developing a one to one intervention
Given the challenges of implementation in the current GCSE resit FE context, an alternative
is to consider the development of a one to one intervention based on the RME number
materials. There is evidence for the effectiveness of one to one tutoring in mathematics in
other phases7 . As a supplement to regular instruction, such an approach would overcome
some of the contextual issues particularly around attendance. However, If RME was the
basis for one to one tutoring an issue would arise about potential conflicts between RME
approaches and regular class teaching. So this would not necessarily reduce the need for a
programme of teacher CPD.
8.5.6.1.3 Professional development needs
For an intervention of either design to be scalable, consideration would need to be given to
how the intervention could be implemented by others and so what would constitute
effective and achievable CPD for teachers.
Whilst the implementation was limited and so strong conclusions cannot be drawn there are
some suggestions of issues to consider. Two of the teachers had worked for a number of
years with MMU researchers. In one case the teacher described how this had influenced her
teaching, particularly using the two models. In the other case, the teacher was honest that
whilst he was positive about the models and methods, pressures to focus on the exam
meant that the impact on his own teaching had been limited. This underlines the challenges
of changing teachers’ practice. Further, when asked to describe the important features of
RME, all four teachers focused on the bar model with one also mentioning the ratio table.
This means that the underlying principles of RME were possibly not that well understood.
7 Torgerson, C., Wiggins, A., Torgerson, D., Ainsworth, H., & Hewitt, C. (2013). Every Child Counts:
testing policy effectiveness using a randomised controlled trial, designed, conducted and reported to
CONSORT standards. Research in Mathematics Education, 15(2), 141-153.
141
However, it is important to note that teachers had not been actively engaged in a
professional development project but had only largely observed RME teaching and had
informal discussions with researchers. Although evidence from these teachers must be
treated with caution, it suggests that observation alone, for example through use of video
CPD materials, will not be sufficient for post-16 teachers to adopt RME approaches. RME
professional development in the post-16 sector will require considerable investment.
8.5.6.2 The prospects for trialling future interventions
Here we provide indications of the size and scope of trials needed for testing future
developments of the intervention. Given the need for further development and the
outcomes of analysis presented in this report, it is not appropriate to calculate minimum
detectable effect size at this point; rather indications of the size of trial needed are given.
Before moving to a full trial, further piloting of redesigned interventions would be required,
unless other relevant evidence of potential effect size is available.
8.5.6.2.1 Whole class intervention
A whole class intervention could be evaluated using a clustered randomised controlled trial.
A useful reference point is the EEF approach to security rating8. To have a clustered trial
with a minimum rating of 4/5 in security in terms of detectable effect size, then there might
typically be 40-50 sites with two classes per site in both the intervention and control
conditions - in the order of 2000 students in each condition. One centre that took part in the
RME GCSE resit project had 400 students annually in GCSE resit classes. However, the issue
of clustering in a trial means that increasing the number of students on a particular site
would be less important in terms of the power of a trial than increasing the number of sites.
It would be important to take steps to control attrition, a particular issue in the FE sector
and it would be prudent to over recruit in the expectation of attrition.
The number of FE and sixth form colleges combined is approximately 300 centres and these
might be the sites which would consider themselves as having the greatest need in relation
to GCSE resits. There are approximately a further 1500 state schools that have post-16
provision. However, the size of GCSE resit groups in these centres is probably smaller and
recruitment to a trial may be more challenging. Recruiting a total of 100 sites, and for
validity a large number of these being FE sites, would be difficult.
8
https://v1.educationendowmentfoundation.org.uk/uploads/pdf/Classifying_the_security_of_EEF_findings_FINAL.pdf
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For a trial of this size, delivery would need at least in part to be by class teachers and so
requires the development of an appropriate professional development programme and
appropriate piloting of these.
8.5.6.2.2 One to one interventions
Designing and implementing a one to one intervention should be considered if a credible
theory of change mechanism warrants it, including potentially evidence of success of such
approaches in other contexts. If such an approach was considered then an advantage of one
to one interventions is that recruitment and randomisation can happen at individual level. A
consequence of this is that the numbers of students in the trial can be much lower, for
example an evaluation of Every Child Counts had 600 students in the trial9. Detection of an
effect size of the order of 0.2 would require a multisite trial involving 15-20 sites with 20- 30
students randomised to the intervention and control condition who then individually
received an RME intervention on a one to one basis. Ideally, such a trial would include a
placebo - potentially of additional mathematics tuition in classes for an equivalent amount
of time to the one to one intervention.
8.5.6.2.3 Implementation issues
The RME GCSE resit project and this evaluation has identified that implementation in the FE
context is challenging as is maintaining robust and secure protocols for testing. Independent
invigilation would need to be included in any future trial costs. As would clear memoranda
of understanding with participants about what is expected for participation.
8.5.7 Conclusion
8.5.7.1 Limitations
Sample size and attrition. There were 147 participants in the study, across intervention and
control conditions. However, complete data sets were obtained for only 52 (29 in the
intervention group and 23 controls). This is a very high rate of attrition, reflecting the
volatility within this age-group and level of study with regard to attendance and
engagement. Attrition poses a considerable risk to validity, as it is not possible to dissociate
reasons for withdrawal from factors relating to the intervention or to assessment. This is
perhaps the most important limitation for future researchers of post-16 mathematics
9 Torgerson, C., Wiggins, A., Torgerson, D., Ainsworth, H., & Hewitt, C. (2013). Every Child Counts: testing policy
effectiveness using a randomised controlled trial, designed, conducted and reported to CONSORT standards.
Research in Mathematics Education, 15(2), 141-153.
143
interventions with this population to consider; researchers must consider ways in which
retention of participants could be improved.
Allocation to conditions. Participants were not allocated to conditions at random. This has
implications for potential bias, and for balance (e.g. participants in the control condition
performed better on the Number pre-test than participants in the intervention condition).
Ideally, allocation to conditions should be carried out either at random, or, following pre-
test in such a manner that there is balance in key variables between conditions.
Test design. Tests for the assessment of number, algebra and other topics in mathematics
were designed by the MMU team responsible for delivering the intervention. This brings
risks that the measures are not sufficiently independent from the intervention10. Use of
independently developed tests could have improved the validity of the evaluation. An
alternative approach could have been to carry out a process evaluation with control groups
in order to confirm that equivalent content had been covered in control group sessions as
had been covered in intervention group sessions.
Test administration. There was inconsistency between sessions and between conditions in
the administration of some tests. All tests should be carried out under the same conditions,
ideally with independent invigilation in order to ensure that scores for each participant are
independent from one another and the class teacher.
8.5.7.2 Conclusions
The RME GCSE Resit intervention led to a short term impact on number but did not lead to
sustained impact on number. There were indications that the MMU RME approach - at least
in the case of the number module the use of the specific models taught - is potentially
beneficial. However, the intervention was relatively brief with only a maximum of 12 hours
of alternative teaching for number and 9 for algebra using the RME approach, with some
students receiving less than this, particularly in the case of algebra. Therefore, it is not
possible to draw conclusions about the potential of RME if it was embedded into a full GCSE
resit course or in general about the value of RME for this age group or profile of learners.
The intervention has contributed to knowledge about the context and needs of learners in
the post-16 contexts. This can support future intervention design.
10 Slavin, R., & Madden, N. A. (2011). Measures inherent to treatments in program effectiveness reviews.
Journal of Research on Educational Effectiveness, 4(4), 370-380.
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The evaluation and study underline the need for developing interventions to address the
needs of GCSE resit students. However, they also indicate how challenging it is to address
these needs and further to evidence success in this area through an experimental approach.
Faculty of Education Manchester Metropolitan University Brooks Building 53 Bonsall Street Manchester M15 6GX
Email: [email protected] Website: mmu.ac.uk/education
MMUEducation
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